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Research ArticleVague Filters of Residuated Lattices
Shokoofeh Ghorbani
Department of Mathematics 22 Bahman Boulevard Kerman 76169-133 Iran
Correspondence should be addressed to Shokoofeh Ghorbani shghorbaniukacir
Received 2 May 2014 Revised 7 August 2014 Accepted 13 August 2014 Published 10 September 2014
Academic Editor Hong J Lai
Copyright copy 2014 Shokoofeh GhorbaniThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Notions of vague filters subpositive implicative vague filters and Boolean vague filters of a residuated lattice are introduced andsome related properties are investigated The characterizations of (subpositive implicative Boolean) vague filters is obtained Weprove that the set of all vague filters of a residuated lattice forms a complete lattice andwe find its distributive sublatticesThe relationamong subpositive implicative vague filters and Boolean vague filters are obtained and it is proved that subpositive implicative vaguefilters are equivalent to Boolean vague filters
1 Introduction
In the classical set there are only two possibilities for anyelements in or not in the set Hence the values of elementsin a set are only one of 0 and 1 Therefore this theory cannothandle the data with ambiguity and uncertainty
Zadeh introduced fuzzy set theory in 1965 [1] to handlesuch ambiguity and uncertainty by generalizing the notionof membership in a set In a fuzzy set 119860 each element 119909 isassociated with a point-value 120583
119860(119909) selected from the unit
interval [0 1] which is termed the grade of membership inthe set This membership degree contains the evidences forboth supporting and opposing 119909
A number of generalizations of Zadehrsquos fuzzy set theoryare intuitionistic fuzzy theory L-fuzzy theory and vaguetheory Gau and Buehrer proposed the concept of vagueset in 1993 [2] by replacing the value of an element in aset with a subinterval of [0 1] Namely a true membershipfunction 119905V(119909) and a false-membership function 119891V(119909) areused to describe the boundaries of membership degreeThese two boundaries form a subinterval [119905V(119909) 1 minus 119891V(119909)]
of [0 1] The vague set theory improves description of theobjective real world becoming a promising tool to deal withinexact uncertain or vague knowledge Many researchershave applied this theory to many situations such as fuzzycontrol decision-making knowledge discovery and faultdiagnosis Recently in [3] Jun andPark introduced the notion
of vague ideal in pseudo MV-algebras and Broumand Saeid[4] introduced the notion of vague BCKBCI-algebras
The concept of residuated latticeswas introduced byWardand Dilworth [5] as a generalization of the structure of theset of ideals of a ring These algebras are a common structureamong algebras associated with logical systems (see [6ndash9])The residuated lattices have interesting algebraic and logicalproperties The main example of residuated lattices relatedto logic is and BL-algebras A basic logic algebra (BL-algebrafor short) is an important class of logical algebras introducedby Hajek [10] in order to provide an algebraic proof of thecompleteness of ldquoBasic Logicrdquo (BL for short) Continuous t-norm based fuzzy logics have been widely applied to fuzzymathematics artificial intelligence and other areasThe filtertheory plays an important role in studying these logicalalgebras and many authors discussed the notion of filters ofthese logical algebras (see [11 12]) From a logical point ofview a filter corresponds to a set of provable formulas
In this paper the concept of vague sets is applied to resid-uated lattices In Section 2 some basic definitions and resultsare explained In Section 3 we introduce the notion of vaguefilters of a residuated lattice and investigate some relatedproperties Also we obtain the characterizations of vaguefilters In Section 4 the smallest vague filters containing agiven vague set are established and we obtain the distributivesublattice of the set of all vague filters of a residuated latticeIn Section 5 notions of vague filters subpositive implicative
Hindawi Publishing CorporationJournal of Discrete MathematicsVolume 2014 Article ID 120342 9 pageshttpdxdoiorg1011552014120342
2 Journal of Discrete Mathematics
vague filters and Boolean vague filters of a residuated latticeare introduced and some related properties are obtained
2 Preliminaries
We recall some definitions and theorems which will beneeded in this paper
Definition 1 (see [6]) A residuated lattice is an algebraicstructure (119871 and or lowast rarr 999492999484 0 1) such that
(1) (119871 and or 0 1) is a bounded lattice with the least ele-ment 0 and the greatest element 1
(2) (119871 lowast 1) is amonoid that islowast is associative and119909lowast1 =
1 lowast 119909 = 119909
(3) 119909 lowast 119910 le 119911 if and only if 119909 le 119910 rarr 119911 if and only if119910 le 119909 999492999484 119911 for all 119909 119910 119911 isin 119871
In the rest of this paper we denote the residuated lattice(119871 and or lowast rarr 999492999484 0 1) by 119871 not119909 = 119909 rarr 0 and sim 119909 = 119909 999492999484 0
Proposition 2 (see [6 9]) Let 119871 be a residuated lattice Thenwe have the following properties for all 119909 119910 119911 isin 119871
Definition 7 (see [1]) Let119883 be a nonempty set A fuzzy set in119883 is a mapping 120583 119883 rarr [0 1]
Proposition 8 (see [12]) Let 119871 be a residuated lattice A fuzzyset 120583 is called a fuzzy filter in 119871 if it satisfies
(fF1) 119909 le 119910 imply 120583(119909) le 120583(119910)
( fF2) min120583(119909) 120583(119910) le 120583(119909 lowast 119910)
for all 119909 119910 isin 119871
Definition 9 (see [13]) A vague set 119860 in the universe ofdiscourse 119883 is characterized by two membership functionsgiven by
(i) a true membership function 119905119860
119883 rarr [0 1]
(ii) a false membership function 119891119860
119883 rarr [0 1]
where 119905119860(119909) is a lower bound on the grade of membership of
119883 derived from the ldquoevidence for 119909rdquo 119891119860(119909) is a lower bound
on the negation of 119909 derived from the ldquoevidence against 119909rdquoand 119905119860(119909) + 119891
119860(119909) le 1
Thus the grade of membership of 119909 in the vague set 119860
is bounded by a subinterval [119905119860(119909) 1 minus 119891
119860(119909)] of [0 1] This
indicates that if the actual grade of membership of 119909 is 120583(119909)then 119905
119860(119909) le 120583(119909) le 1 minus 119891
119860(119909) The vague set 119860 is written
as 119860 = ⟨119909 [119905119860(119909) 1 minus 119891
119860(119909)]⟩ 119909 isin 119883 where the interval
[119905119860(119909) 1 minus 119891
119860(119909)] is called the vague value of 119909 in 119860 denoted
by 119881119860(119909)
Definition 10 (see [13]) A vague set 119860 of a set 119883 is called
(1) the zero vague set of 119883 if 119905119860(119909) = 0 and 119891
119860(119909) = 1 for
all 119909 isin 119883
(2) the unit vague set of 119883 if 119905119860(119909) = 1 and 119891
119860(119909) = 0 for
all 119909 isin 119883
(3) the 120572-vague set of 119883 if 119905119860(119909) = 120572 and 119891
119860(119909) = 1 minus 120572
for all 119909 isin 119883 where 120572 isin (0 1)
Let 119868[0 1] denote the family of all closed subintervals of[0 1] Nowwedefine the refinedminimum (briefly imin) and
Journal of Discrete Mathematics 3
an order le on elements 1198681= [1198861 1198871] and 119868
2= [1198862 1198872] of 119868[0 1]
asimin (119868
1 1198682) = [min 119886
1 1198862 min 119887
1 1198872]
1198681le 1198682
iff 1198861le 1198862 1198871le 1198872
(1)
Similarly we can define ge = and imax Then the concept ofimin and imax could be extended to define iinf and isup ofinfinite number of elements of 119868[0 1] It is a known fact that119871 = 119868[0 1] iinf isup le is a lattice with universal boundszero vague set and unit vague set For 120572 120573 isin [0 1] we nowdefine (120572 120573)-cut and 120572-cut of a vague set
Definition 11 (see [13]) Let 119860 be a vague set of a universe 119883
with the true-membership function 119905119860and false-membership
function119891119860The (120572 120573)-cut of the vague set119860 is a crisp subset
119860(120572120573)
of the set 119883 given by
119860(120572120573)
= 119909 isin 119883 119881119860 (119909) ge [120572 120573]
where 120572 le 120573
(2)
Clearly 119860(00)
= 119883 The (120572 120573)-cuts are also called vague-cutsof the vague set 119860
Definition 12 (see [13]) The 120572-cut of the vague set119860 is a crispsubset 119860
120572of the set119883 given by 119860
120572= 119860(120572120572)
Equivalently wecan define the 120572-cut as 119860
120572= 119909 isin 119883 119905
119860(119909) ge 120572
3 Vague Filters of Residuated Lattices
In this section we define the notion of vague filters ofresiduated lattices and obtain some related results
Definition 13 A vague set 119860 of a residuated lattice 119871 is calleda vague filter of 119871 if the following conditions hold
(VF1) if 119909 le 119910 then 119881119860(119909) le 119881
119860(119910) for all 119909 119910 isin 119871
(VF2) imin119881119860(119909) 119881119860(119910) le 119881
119860(119909lowast119910) for all 119909 119910 isin
119871
That is(1) 119909 le 119910 implies 119905
119860(119909) le 119905
119860(119910) and 1minus119891
119860(119909) le 1minus119891
119860(119910)
for all 119909 119910 isin 119871(2) min119905
119860(119909) 119905119860(119910) le 119905
119860(119909 lowast 119910) and min1 minus 119891
119860(119909) 1 minus
119891119860(119910) le 1 minus 119891
119860(119909 lowast 119910) for all 119909 119910 isin 119871
In the following proposition we will show that the vaguefilters of a residuated lattice exist
Proposition 14 Unit vague set and120572-vague set of a residuatedlattice 119871 are vague filters of 119871
Proof Let 119860 be a 120572-vague set of 119871 It is clear that 119860 satisfies(VF1) For 119909 119910 isin 119871 we have
119905119860(119909 lowast 119910) = 120572 = min 120572 120572 = min 119905
119860 (119909) 119905119860 (119910)
1 minus 119891119860(119909 lowast 119910)
= 120572 = min 120572 120572 = min 1 minus 119891119860 (119909) 1 minus 119891
119860(119910)
(3)
Thus imin119881119860(119909) 119881119860(119910) le 119881
119860(119909 lowast 119910) for all 119909 119910 isin 119871 Hence
it is a vague filter of 119871 The proof of the other cases is similar
Lemma 15 Let 119860 be a vague set of a residuated lattice 119871 suchthat it satisfies (VF1) Then the following are equivalent
(VF4) iminVA(x)VA(x rarr y) le VA(y)(VF5) iminVA(x)VA(x 999492999484 y) le VA(y)
for all 119909 119910 isin 119871
Proof Suppose that 119860 satisfies (VF4) We have 119909 le (119909 999492999484
119910) rarr 119910 By (VF1) 119881119860(119909) le 119881
119860((119909 999492999484 119910) rarr 119910) We get that
ge imin 119881119860 (119909) 119881119860 (119910)
(8)
Hence (VF2) holds
Theorem 17 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter if and only if it satisfies (VF5) and(VF6)
4 Journal of Discrete Mathematics
Definition 18 A vague set 119860 of a residuated lattice 119871 is calleda vague lattice filter of 119871 if it satisfies (VF1) and
(VLF2) imin119881119860(119909) 119881119860(119910) le 119881
119860(119909 and 119910) for all
119909 119910 isin 119871
Proposition 19 Let 119860 be a vague filter of a residuated lattice119871 then 119860 is a vague lattice filter
In the following example we will show that the converseof Proposition 19 may not be true in general
Example 20 Let 0 119886 119887 119888 1 such that 0 lt 119886 lt 119887 119888 lt 1where 119887 and 119888 are incomparable Define the operations lowast rarr and 999492999484 as follows
lowast 0 119886 119887 119888 1
0 0 0 0 0 0
119886 0 0 0 119886 119886
119887 0 119886 119887 119886 119887
119888 0 0 0 119888 119888
1 0 119886 119887 119888 1
997888rarr 0 119886 119887 119888 1
0 1 1 1 1 1
119886 119888 1 1 1 1
119887 119888 119888 1 119888 1
119888 0 119887 119887 1 1
1 0 119886 119887 119888 1
999492999484 0 119886 119887 119888 1
0 1 1 1 1 1
119886 119887 1 1 1 1
119887 0 119888 1 119888 1
119888 119887 119887 119887 1 1
1 0 119886 119887 119888 1
(9)
Then 119871 is a residuated lattice Let 119860 be the vague set definedas follows
In the following theorems we will study the relationshipbetween filters and vague filters of a residuated lattice
Theorem 21 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if for all 120572 120573 isin [0 1]the set 119860
(120572120573)is either empty or a filter of 119871 where 120572 le 120573
Proof Suppose that 119860 is a vague filter of 119871 and 120572 120573 isin [0 1]Let 119860
(120572120573)= 0 We will show that 119860
(120572120573)is a filter of 119871
(1) Suppose that 119909 le 119910 and 119909 isin 119860(120572120573)
By (VF1)119881119860(119910) ge
119881119860(119909) ge [120572 120573] Therefor 119910 isin 119860
(120572120573)
(2) Suppose that 119909 119910 isin 119860(120572120573)
Then 119881119860(119909) 119881119860(119910) ge
[120572 120573] By (VF2) we have 119881119860(119909 lowast 119910) ge
Hence (VF2) holdsSuppose that 119909 le 119910 119905
119860(119909) = 120572 and 1 minus 119891
119860(119909) = 120573 Hence
119909 isin 119860(120572120573)
By assumption 119860(120572120573)
is a filter We get that 119910 isin
119860(120572120573)
that is 119881119860(119910) ge [120572 120573] = [119905
119860(119909) 1 minus 119891
119860(119909)] = 119881
119860(119909)
Hence 119860 is a vague filter of 119871
The filters like119860(120572120573)
are also called vague-cuts filters of 119871
Corollary 22 Let 119860 be a vague filter of a residuated lattice 119871Then for all 120572 isin [0 1] the set 119860
120572is either empty or a filter of
119871
Corollary 23 Any filter 119865 of a residuated lattice 119871 is a vague-cut filter of some vague filter of 119871
Proof For all 119909 isin 119871 define
1 minus 119891119860 (119909) = 119905
119860 (119909) = 1 if 119909 isin 119865
120572 otherwise(13)
Hence we have 119881119860(119909) = [1 1] if 119909 isin 119865 and 119881
119860(119909) = [120572 120572]
otherwise where 120572 isin (0 1) It is clear that 119865 = 119860(11)
Let119909 119910 isin 119871 If 119909 119910 isin 119865 then 119909 lowast 119910 isin 119865 We have 119881
119860(119909 lowast 119910) =
[1 1] = imin119881119860(119909) 119881119860(119910) If one of 119909 or 119910 does not belong
to 119865 then one of119881119860(119909) and119881
119860(119910) is equal to [120572 120572]Therefore
119881119860(119909lowast119910) ge [120572 120572] = imin119881
119860(119909) 119881119860(119910) Suppose that 119909 le 119910
If 119909 isin 119865 then 119910 isin 119865 Hence 119881119860(119909) = 119881
119860(119910) If 119909 notin 119865 then
119881119860(119909) = [120572 120572] We have 119881
119860(119909) le 119881
119860(119910) Hence 119860 is a vague
filter of 119871
In the following proposition we will show that vaguefilters of a residuated lattice are a generalization of fuzzyfilters
Proposition 24 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if the fuzzy sets 119905
119860and
1 minus 119891119860are fuzzy filters in 119871
Journal of Discrete Mathematics 5
4 Lattice of Vague Filters
Let 119860 and 119861 be vague sets of a residuated lattice 119871 If 119881119860(119909) le
119881119861(119909) for all 119909 isin 119871 then we say 119861 containing119860 and denote it
by 119860 ⪯ 119861The unite vague set of a residuated lattice 119871 is a vague
filter of 119871 containing any vague filter of 119871 For a vague setwe can define the intersection of two vague sets 119860 and 119861 of aresiduated lattice 119871 by 119881
119860cap119861(119909) = imin119881
119860(119909) 119881119861(119909) for all
119909 isin 119871 It is easy to prove that the intersection of any family ofvague filters of 119871 is a vague filter of 119871 Hence we can define avague filter generated by a vague set as follows
Definition 25 Let 119860 be a vague set of a residuated lattice 119871A vague filter 119861 is called a vague filter generated by 119860 if itsatisfies
(i) 119860 ⪯ 119861(ii) 119860 ⪯ 119862 that implies 119861 ⪯ 119862 for all vague filters 119862 of 119871
The vague filter generated by 119860 is denoted by ⟨119860⟩
Example 26 Consider the vague lattice filter119860 in Example 20which is not a vague filter of 119871Then the vague filter generatedby 119860 is
⟨119860⟩ = ⟨0 [03 04]⟩ ⟨119886 [03 04]⟩
⟨119887 [03 04]⟩ ⟨119888 [03 08]⟩ ⟨0 [05 09]⟩
(14)
Theorem 27 Let 119860 be a vague set of a residuated lattice 119871Then the vague value of 119909 in ⟨119860⟩ is
119881⟨119860⟩ (119909) = isup imin VA (a1) VA (a2) VA (an)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
(15)
for all 119909 isin 119871
Proof First we will prove that ⟨119860⟩ is a vague set of 119871
119905⟨119860⟩ (119909)
= sup min 119905119860(1198861) 119905119860(1198862) 119905
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
le sup min 1 minus 119891119860(1198861) 1 minus 119891
119860(1198862) 1 minus 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= sup 1 minus max 119891119860(1198861) 119891119860(1198862) 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 1 minus inf max 119891119860(1198861) 119891119860(1198862) 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 1 minus 119891⟨119860⟩ (119909)
(16)
Now we will show that ⟨119860⟩ is a vague filter of 119871(VF1) Suppose that 119909 119910 isin 119871 is arbitrary For every 120576 gt 0
we can take 1198861 119886
119899 1198871 119887
119898isin 119871 such that119909 ge 119886
1lowastsdot sdot sdotlowast119886
119899
119910 ge 1198871lowast sdot sdot sdot lowast 119887
119898 and
119905⟨119860⟩ (119909) minus 120576 lt min 119905
119860(1198861) 119905119860(1198862) 119905
119860(119886119899)
119905⟨119860⟩
(119910) minus 120576 lt min 119905119860(1198871) 119905119860(1198872) 119905
119860(119887119898)
(17)
By Proposition 2 part (4) we have 119909 lowast 119910 ge 1198861lowast sdot sdot sdot lowast 119886
119899lowast 1198871lowast
sdot sdot sdot lowast 119887119898and then
119905⟨119860⟩
(119909 lowast 119910)
ge min 119905119860(1198861) 119905
119860(119886119899) 119905119860(1198871) 119905
119860(119887119898)
= min min 119905119860(1198861) 119905
119860(119886119899)
min 119905119860(1198871) 119905
119860(119887119898)
gt min (119905⟨119860⟩ (119909) minus 120576) (119905
⟨119860⟩(119910) minus 120576)
= min 119905⟨119860⟩ (119909) 119905⟨119860⟩ (119910) minus 120576
(18)
Since 120576 gt 0 is arbitrary we obtain that 119905⟨119860⟩
(119909 lowast 119910) ge
min119905⟨119860⟩
(119909) 119905⟨119860⟩
(119910)Similarly we can show that 1 minus 119891
⟨119860⟩(119909 lowast 119910) ge min1 minus
119891⟨119860⟩
(119909) 1 minus119891⟨119860⟩
(119910) Hence119881119860(119909lowast119910) ge imin119881
119860(119909) 119881119860(119910)
for all 119909 119910 isin 119871(VF2) Suppose that 119909 le 119910where 119909 119910 isin 119871 Let 119886
1 119886
119899isin
119871 such that 1198861lowast sdot sdot sdot lowast 119886
119899le 119909 Then 119886
1lowast sdot sdot sdot lowast 119886
119899le 119910 And we
get that
119881119860(119910) ge imin 119881
119860(1198861) 119881119860(1198862) 119881
119860(119886119899) (19)
Since 1198861 119886
119899isin 119871 where 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899is arbitrary then
119881⟨119860⟩
(119910) ge isup imin 119881119860(1198861) 119881119860(1198862) 119881
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 119881⟨119860⟩ (119909)
(20)
Therefore ⟨119860⟩ is a vague filter of 119871 by Theorem 16 It is clearthat119860 ⪯ ⟨119860⟩ Now let119862 be a vague filter of119871 such that119860 ⪯ 119862Then we have
119881⟨119860⟩ (119909)
= isup imin 119881119860(1198861) 119881
119860(119886119899) 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899
le isup imin 119881119862(1198861) 119881
119862(119886119899) 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899
= 119881119862 (119909)
(21)
for all 119909 isin 119871 Hence ⟨119860⟩ ⪯ 119862 and then ⟨119860⟩ is a vague filtergenerated by 119860
We denote the set of all vague filters of a residuated lattice119871 by VF(119871)
6 Journal of Discrete Mathematics
Corollary 28 (VF(119871) cap or) is a complete lattice
Proof Clearly if 119860119894119894isinΓ
is a family of vague filters of aresiduated lattice 119871 then the infimumof this family is⋂
119894isinΓ119860119894
and the supremum is ⋁119894isinΓ
119860119894= ⟨⋃119894isinΓ
119860119894⟩ It is easy to prove
that (IFF(119871) cap or) is a complete lattice
Let 119860 and 119861 be vague sets of a residuated lattice 119871 Then119860 lowast 119861 is defined as follows
119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910
ge imin 119881119860 (119911) 119881119861 (1) ge 119881
119860 (119911)
(26)
Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then
Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881
119862(119911) by (VF2)
Hence 119881119862(119911) ge isup119881
119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have
1 lowast 119911 = 119911 We obtain that
119881119862 (119911) = 119881
119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)
(2) It follows from Proposition 2 part (2) andDefinition 13
Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881
119860(1) = 119881
119861(1)
Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice
Journal of Discrete Mathematics 7
Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32
119860((119909 999492999484 119910) rarr 119910) for any
119909 119910 119911 isin 119871
Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies
119860((119909 rarr 119910) 999492999484 119910) ge
[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can
prove that 119860 satisfies (SVF4)
In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter
Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881
119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484
119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119910) 999492999484 119910)]) 119881119860(1) le 119881
119860((119909 rarr 119910) 999492999484 119910) Similarly we
can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter
Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
119860((119910 rarr 119911) 999492999484 119910) le 119881
119860((119910 rarr 119911) 999492999484
8 Journal of Discrete Mathematics
((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119911 999492999484 119910) Since 119860 is a vague
filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881
119860(119911 999492999484 119910) 119881
119860((119911 999492999484
119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously
we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have
119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we
can prove that (SVF8) holds
Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)
Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119910) Hence 119860 satisfies (SVF5)
Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38
Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF11) 119881119860(not119909 999492999484 119909) le 119881
119860(119909) for any 119909 isin 119871
(SVF12) 119881119860(sim 119909 rarr 119909) le 119881
119860(119909) for any 119909 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence
(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881
119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(not119909 999492999484 119909) le 119881
119860(119909)
Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40
The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem
Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881
119860(1) = 119881
119861(1) If 119860 is a
subpositive implicative vague filter of L so 119861 is
Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then
119909)) le 119881119860(119906 rarr 119909) Hence 119881
119860(119906 rarr 119909) = 119881
119860(1) = 119881
119861(1)
Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881
119860(119906 rarr 119909) = 119881
119861(1)
We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881
119861(119906) = imin119881
119861(119906 rarr
119909) 119881119861(119906) le 119881
119861(119909) Similarly we can show that 119866 satisfies
(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40
Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if
(BVF1) 119881119860(119909 or not119909) = 119881
119860(1) for all 119909 isin 119871
(BVF2) 119881119860(119909or sim 119909) = 119881
119860(1) for all 119909 isin 119871
Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters
Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860
(120572120573)is either empty or a Boolean filter of
119871 where 120572 le 120573
Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter
Proof Let119860 be a subpositive implicative vague filterWe have
119881119860 (1) = 119881
119860 (not (119909 or not119909) 999492999484 (119909 or not119909))
= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881
119860 (119909 or not119909)
(33)
Hence 119881119860(119909 or not119909) = 119881
119860(1) Similarly we can prove that 119860
satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881
119860(119909ornot119909) = 119881
119860(1)We have119881
119860(119909ornot119909) =
119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881
119860((119909 or not119909) 999492999484 119909) = imin119881
119860((119909 or not119909) 999492999484 119909) 119881
119860(119909 or not119909) le
119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence
119860 is a subpositive implicative vague filter byTheorem 41
Journal of Discrete Mathematics 9
6 Conclusion
In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters
Conflict of Interests
There is no conflict of interests regarding the publication ofthis paper
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009
[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009
[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972
[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001
[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007
[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends
in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998
[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011
[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010
[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006
vague filters and Boolean vague filters of a residuated latticeare introduced and some related properties are obtained
2 Preliminaries
We recall some definitions and theorems which will beneeded in this paper
Definition 1 (see [6]) A residuated lattice is an algebraicstructure (119871 and or lowast rarr 999492999484 0 1) such that
(1) (119871 and or 0 1) is a bounded lattice with the least ele-ment 0 and the greatest element 1
(2) (119871 lowast 1) is amonoid that islowast is associative and119909lowast1 =
1 lowast 119909 = 119909
(3) 119909 lowast 119910 le 119911 if and only if 119909 le 119910 rarr 119911 if and only if119910 le 119909 999492999484 119911 for all 119909 119910 119911 isin 119871
In the rest of this paper we denote the residuated lattice(119871 and or lowast rarr 999492999484 0 1) by 119871 not119909 = 119909 rarr 0 and sim 119909 = 119909 999492999484 0
Proposition 2 (see [6 9]) Let 119871 be a residuated lattice Thenwe have the following properties for all 119909 119910 119911 isin 119871
Definition 7 (see [1]) Let119883 be a nonempty set A fuzzy set in119883 is a mapping 120583 119883 rarr [0 1]
Proposition 8 (see [12]) Let 119871 be a residuated lattice A fuzzyset 120583 is called a fuzzy filter in 119871 if it satisfies
(fF1) 119909 le 119910 imply 120583(119909) le 120583(119910)
( fF2) min120583(119909) 120583(119910) le 120583(119909 lowast 119910)
for all 119909 119910 isin 119871
Definition 9 (see [13]) A vague set 119860 in the universe ofdiscourse 119883 is characterized by two membership functionsgiven by
(i) a true membership function 119905119860
119883 rarr [0 1]
(ii) a false membership function 119891119860
119883 rarr [0 1]
where 119905119860(119909) is a lower bound on the grade of membership of
119883 derived from the ldquoevidence for 119909rdquo 119891119860(119909) is a lower bound
on the negation of 119909 derived from the ldquoevidence against 119909rdquoand 119905119860(119909) + 119891
119860(119909) le 1
Thus the grade of membership of 119909 in the vague set 119860
is bounded by a subinterval [119905119860(119909) 1 minus 119891
119860(119909)] of [0 1] This
indicates that if the actual grade of membership of 119909 is 120583(119909)then 119905
119860(119909) le 120583(119909) le 1 minus 119891
119860(119909) The vague set 119860 is written
as 119860 = ⟨119909 [119905119860(119909) 1 minus 119891
119860(119909)]⟩ 119909 isin 119883 where the interval
[119905119860(119909) 1 minus 119891
119860(119909)] is called the vague value of 119909 in 119860 denoted
by 119881119860(119909)
Definition 10 (see [13]) A vague set 119860 of a set 119883 is called
(1) the zero vague set of 119883 if 119905119860(119909) = 0 and 119891
119860(119909) = 1 for
all 119909 isin 119883
(2) the unit vague set of 119883 if 119905119860(119909) = 1 and 119891
119860(119909) = 0 for
all 119909 isin 119883
(3) the 120572-vague set of 119883 if 119905119860(119909) = 120572 and 119891
119860(119909) = 1 minus 120572
for all 119909 isin 119883 where 120572 isin (0 1)
Let 119868[0 1] denote the family of all closed subintervals of[0 1] Nowwedefine the refinedminimum (briefly imin) and
Journal of Discrete Mathematics 3
an order le on elements 1198681= [1198861 1198871] and 119868
2= [1198862 1198872] of 119868[0 1]
asimin (119868
1 1198682) = [min 119886
1 1198862 min 119887
1 1198872]
1198681le 1198682
iff 1198861le 1198862 1198871le 1198872
(1)
Similarly we can define ge = and imax Then the concept ofimin and imax could be extended to define iinf and isup ofinfinite number of elements of 119868[0 1] It is a known fact that119871 = 119868[0 1] iinf isup le is a lattice with universal boundszero vague set and unit vague set For 120572 120573 isin [0 1] we nowdefine (120572 120573)-cut and 120572-cut of a vague set
Definition 11 (see [13]) Let 119860 be a vague set of a universe 119883
with the true-membership function 119905119860and false-membership
function119891119860The (120572 120573)-cut of the vague set119860 is a crisp subset
119860(120572120573)
of the set 119883 given by
119860(120572120573)
= 119909 isin 119883 119881119860 (119909) ge [120572 120573]
where 120572 le 120573
(2)
Clearly 119860(00)
= 119883 The (120572 120573)-cuts are also called vague-cutsof the vague set 119860
Definition 12 (see [13]) The 120572-cut of the vague set119860 is a crispsubset 119860
120572of the set119883 given by 119860
120572= 119860(120572120572)
Equivalently wecan define the 120572-cut as 119860
120572= 119909 isin 119883 119905
119860(119909) ge 120572
3 Vague Filters of Residuated Lattices
In this section we define the notion of vague filters ofresiduated lattices and obtain some related results
Definition 13 A vague set 119860 of a residuated lattice 119871 is calleda vague filter of 119871 if the following conditions hold
(VF1) if 119909 le 119910 then 119881119860(119909) le 119881
119860(119910) for all 119909 119910 isin 119871
(VF2) imin119881119860(119909) 119881119860(119910) le 119881
119860(119909lowast119910) for all 119909 119910 isin
119871
That is(1) 119909 le 119910 implies 119905
119860(119909) le 119905
119860(119910) and 1minus119891
119860(119909) le 1minus119891
119860(119910)
for all 119909 119910 isin 119871(2) min119905
119860(119909) 119905119860(119910) le 119905
119860(119909 lowast 119910) and min1 minus 119891
119860(119909) 1 minus
119891119860(119910) le 1 minus 119891
119860(119909 lowast 119910) for all 119909 119910 isin 119871
In the following proposition we will show that the vaguefilters of a residuated lattice exist
Proposition 14 Unit vague set and120572-vague set of a residuatedlattice 119871 are vague filters of 119871
Proof Let 119860 be a 120572-vague set of 119871 It is clear that 119860 satisfies(VF1) For 119909 119910 isin 119871 we have
119905119860(119909 lowast 119910) = 120572 = min 120572 120572 = min 119905
119860 (119909) 119905119860 (119910)
1 minus 119891119860(119909 lowast 119910)
= 120572 = min 120572 120572 = min 1 minus 119891119860 (119909) 1 minus 119891
119860(119910)
(3)
Thus imin119881119860(119909) 119881119860(119910) le 119881
119860(119909 lowast 119910) for all 119909 119910 isin 119871 Hence
it is a vague filter of 119871 The proof of the other cases is similar
Lemma 15 Let 119860 be a vague set of a residuated lattice 119871 suchthat it satisfies (VF1) Then the following are equivalent
(VF4) iminVA(x)VA(x rarr y) le VA(y)(VF5) iminVA(x)VA(x 999492999484 y) le VA(y)
for all 119909 119910 isin 119871
Proof Suppose that 119860 satisfies (VF4) We have 119909 le (119909 999492999484
119910) rarr 119910 By (VF1) 119881119860(119909) le 119881
119860((119909 999492999484 119910) rarr 119910) We get that
ge imin 119881119860 (119909) 119881119860 (119910)
(8)
Hence (VF2) holds
Theorem 17 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter if and only if it satisfies (VF5) and(VF6)
4 Journal of Discrete Mathematics
Definition 18 A vague set 119860 of a residuated lattice 119871 is calleda vague lattice filter of 119871 if it satisfies (VF1) and
(VLF2) imin119881119860(119909) 119881119860(119910) le 119881
119860(119909 and 119910) for all
119909 119910 isin 119871
Proposition 19 Let 119860 be a vague filter of a residuated lattice119871 then 119860 is a vague lattice filter
In the following example we will show that the converseof Proposition 19 may not be true in general
Example 20 Let 0 119886 119887 119888 1 such that 0 lt 119886 lt 119887 119888 lt 1where 119887 and 119888 are incomparable Define the operations lowast rarr and 999492999484 as follows
lowast 0 119886 119887 119888 1
0 0 0 0 0 0
119886 0 0 0 119886 119886
119887 0 119886 119887 119886 119887
119888 0 0 0 119888 119888
1 0 119886 119887 119888 1
997888rarr 0 119886 119887 119888 1
0 1 1 1 1 1
119886 119888 1 1 1 1
119887 119888 119888 1 119888 1
119888 0 119887 119887 1 1
1 0 119886 119887 119888 1
999492999484 0 119886 119887 119888 1
0 1 1 1 1 1
119886 119887 1 1 1 1
119887 0 119888 1 119888 1
119888 119887 119887 119887 1 1
1 0 119886 119887 119888 1
(9)
Then 119871 is a residuated lattice Let 119860 be the vague set definedas follows
In the following theorems we will study the relationshipbetween filters and vague filters of a residuated lattice
Theorem 21 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if for all 120572 120573 isin [0 1]the set 119860
(120572120573)is either empty or a filter of 119871 where 120572 le 120573
Proof Suppose that 119860 is a vague filter of 119871 and 120572 120573 isin [0 1]Let 119860
(120572120573)= 0 We will show that 119860
(120572120573)is a filter of 119871
(1) Suppose that 119909 le 119910 and 119909 isin 119860(120572120573)
By (VF1)119881119860(119910) ge
119881119860(119909) ge [120572 120573] Therefor 119910 isin 119860
(120572120573)
(2) Suppose that 119909 119910 isin 119860(120572120573)
Then 119881119860(119909) 119881119860(119910) ge
[120572 120573] By (VF2) we have 119881119860(119909 lowast 119910) ge
Hence (VF2) holdsSuppose that 119909 le 119910 119905
119860(119909) = 120572 and 1 minus 119891
119860(119909) = 120573 Hence
119909 isin 119860(120572120573)
By assumption 119860(120572120573)
is a filter We get that 119910 isin
119860(120572120573)
that is 119881119860(119910) ge [120572 120573] = [119905
119860(119909) 1 minus 119891
119860(119909)] = 119881
119860(119909)
Hence 119860 is a vague filter of 119871
The filters like119860(120572120573)
are also called vague-cuts filters of 119871
Corollary 22 Let 119860 be a vague filter of a residuated lattice 119871Then for all 120572 isin [0 1] the set 119860
120572is either empty or a filter of
119871
Corollary 23 Any filter 119865 of a residuated lattice 119871 is a vague-cut filter of some vague filter of 119871
Proof For all 119909 isin 119871 define
1 minus 119891119860 (119909) = 119905
119860 (119909) = 1 if 119909 isin 119865
120572 otherwise(13)
Hence we have 119881119860(119909) = [1 1] if 119909 isin 119865 and 119881
119860(119909) = [120572 120572]
otherwise where 120572 isin (0 1) It is clear that 119865 = 119860(11)
Let119909 119910 isin 119871 If 119909 119910 isin 119865 then 119909 lowast 119910 isin 119865 We have 119881
119860(119909 lowast 119910) =
[1 1] = imin119881119860(119909) 119881119860(119910) If one of 119909 or 119910 does not belong
to 119865 then one of119881119860(119909) and119881
119860(119910) is equal to [120572 120572]Therefore
119881119860(119909lowast119910) ge [120572 120572] = imin119881
119860(119909) 119881119860(119910) Suppose that 119909 le 119910
If 119909 isin 119865 then 119910 isin 119865 Hence 119881119860(119909) = 119881
119860(119910) If 119909 notin 119865 then
119881119860(119909) = [120572 120572] We have 119881
119860(119909) le 119881
119860(119910) Hence 119860 is a vague
filter of 119871
In the following proposition we will show that vaguefilters of a residuated lattice are a generalization of fuzzyfilters
Proposition 24 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if the fuzzy sets 119905
119860and
1 minus 119891119860are fuzzy filters in 119871
Journal of Discrete Mathematics 5
4 Lattice of Vague Filters
Let 119860 and 119861 be vague sets of a residuated lattice 119871 If 119881119860(119909) le
119881119861(119909) for all 119909 isin 119871 then we say 119861 containing119860 and denote it
by 119860 ⪯ 119861The unite vague set of a residuated lattice 119871 is a vague
filter of 119871 containing any vague filter of 119871 For a vague setwe can define the intersection of two vague sets 119860 and 119861 of aresiduated lattice 119871 by 119881
119860cap119861(119909) = imin119881
119860(119909) 119881119861(119909) for all
119909 isin 119871 It is easy to prove that the intersection of any family ofvague filters of 119871 is a vague filter of 119871 Hence we can define avague filter generated by a vague set as follows
Definition 25 Let 119860 be a vague set of a residuated lattice 119871A vague filter 119861 is called a vague filter generated by 119860 if itsatisfies
(i) 119860 ⪯ 119861(ii) 119860 ⪯ 119862 that implies 119861 ⪯ 119862 for all vague filters 119862 of 119871
The vague filter generated by 119860 is denoted by ⟨119860⟩
Example 26 Consider the vague lattice filter119860 in Example 20which is not a vague filter of 119871Then the vague filter generatedby 119860 is
⟨119860⟩ = ⟨0 [03 04]⟩ ⟨119886 [03 04]⟩
⟨119887 [03 04]⟩ ⟨119888 [03 08]⟩ ⟨0 [05 09]⟩
(14)
Theorem 27 Let 119860 be a vague set of a residuated lattice 119871Then the vague value of 119909 in ⟨119860⟩ is
119881⟨119860⟩ (119909) = isup imin VA (a1) VA (a2) VA (an)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
(15)
for all 119909 isin 119871
Proof First we will prove that ⟨119860⟩ is a vague set of 119871
119905⟨119860⟩ (119909)
= sup min 119905119860(1198861) 119905119860(1198862) 119905
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
le sup min 1 minus 119891119860(1198861) 1 minus 119891
119860(1198862) 1 minus 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= sup 1 minus max 119891119860(1198861) 119891119860(1198862) 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 1 minus inf max 119891119860(1198861) 119891119860(1198862) 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 1 minus 119891⟨119860⟩ (119909)
(16)
Now we will show that ⟨119860⟩ is a vague filter of 119871(VF1) Suppose that 119909 119910 isin 119871 is arbitrary For every 120576 gt 0
we can take 1198861 119886
119899 1198871 119887
119898isin 119871 such that119909 ge 119886
1lowastsdot sdot sdotlowast119886
119899
119910 ge 1198871lowast sdot sdot sdot lowast 119887
119898 and
119905⟨119860⟩ (119909) minus 120576 lt min 119905
119860(1198861) 119905119860(1198862) 119905
119860(119886119899)
119905⟨119860⟩
(119910) minus 120576 lt min 119905119860(1198871) 119905119860(1198872) 119905
119860(119887119898)
(17)
By Proposition 2 part (4) we have 119909 lowast 119910 ge 1198861lowast sdot sdot sdot lowast 119886
119899lowast 1198871lowast
sdot sdot sdot lowast 119887119898and then
119905⟨119860⟩
(119909 lowast 119910)
ge min 119905119860(1198861) 119905
119860(119886119899) 119905119860(1198871) 119905
119860(119887119898)
= min min 119905119860(1198861) 119905
119860(119886119899)
min 119905119860(1198871) 119905
119860(119887119898)
gt min (119905⟨119860⟩ (119909) minus 120576) (119905
⟨119860⟩(119910) minus 120576)
= min 119905⟨119860⟩ (119909) 119905⟨119860⟩ (119910) minus 120576
(18)
Since 120576 gt 0 is arbitrary we obtain that 119905⟨119860⟩
(119909 lowast 119910) ge
min119905⟨119860⟩
(119909) 119905⟨119860⟩
(119910)Similarly we can show that 1 minus 119891
⟨119860⟩(119909 lowast 119910) ge min1 minus
119891⟨119860⟩
(119909) 1 minus119891⟨119860⟩
(119910) Hence119881119860(119909lowast119910) ge imin119881
119860(119909) 119881119860(119910)
for all 119909 119910 isin 119871(VF2) Suppose that 119909 le 119910where 119909 119910 isin 119871 Let 119886
1 119886
119899isin
119871 such that 1198861lowast sdot sdot sdot lowast 119886
119899le 119909 Then 119886
1lowast sdot sdot sdot lowast 119886
119899le 119910 And we
get that
119881119860(119910) ge imin 119881
119860(1198861) 119881119860(1198862) 119881
119860(119886119899) (19)
Since 1198861 119886
119899isin 119871 where 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899is arbitrary then
119881⟨119860⟩
(119910) ge isup imin 119881119860(1198861) 119881119860(1198862) 119881
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 119881⟨119860⟩ (119909)
(20)
Therefore ⟨119860⟩ is a vague filter of 119871 by Theorem 16 It is clearthat119860 ⪯ ⟨119860⟩ Now let119862 be a vague filter of119871 such that119860 ⪯ 119862Then we have
119881⟨119860⟩ (119909)
= isup imin 119881119860(1198861) 119881
119860(119886119899) 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899
le isup imin 119881119862(1198861) 119881
119862(119886119899) 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899
= 119881119862 (119909)
(21)
for all 119909 isin 119871 Hence ⟨119860⟩ ⪯ 119862 and then ⟨119860⟩ is a vague filtergenerated by 119860
We denote the set of all vague filters of a residuated lattice119871 by VF(119871)
6 Journal of Discrete Mathematics
Corollary 28 (VF(119871) cap or) is a complete lattice
Proof Clearly if 119860119894119894isinΓ
is a family of vague filters of aresiduated lattice 119871 then the infimumof this family is⋂
119894isinΓ119860119894
and the supremum is ⋁119894isinΓ
119860119894= ⟨⋃119894isinΓ
119860119894⟩ It is easy to prove
that (IFF(119871) cap or) is a complete lattice
Let 119860 and 119861 be vague sets of a residuated lattice 119871 Then119860 lowast 119861 is defined as follows
119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910
ge imin 119881119860 (119911) 119881119861 (1) ge 119881
119860 (119911)
(26)
Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then
Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881
119862(119911) by (VF2)
Hence 119881119862(119911) ge isup119881
119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have
1 lowast 119911 = 119911 We obtain that
119881119862 (119911) = 119881
119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)
(2) It follows from Proposition 2 part (2) andDefinition 13
Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881
119860(1) = 119881
119861(1)
Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice
Journal of Discrete Mathematics 7
Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32
119860((119909 999492999484 119910) rarr 119910) for any
119909 119910 119911 isin 119871
Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies
119860((119909 rarr 119910) 999492999484 119910) ge
[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can
prove that 119860 satisfies (SVF4)
In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter
Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881
119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484
119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119910) 999492999484 119910)]) 119881119860(1) le 119881
119860((119909 rarr 119910) 999492999484 119910) Similarly we
can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter
Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
119860((119910 rarr 119911) 999492999484 119910) le 119881
119860((119910 rarr 119911) 999492999484
8 Journal of Discrete Mathematics
((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119911 999492999484 119910) Since 119860 is a vague
filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881
119860(119911 999492999484 119910) 119881
119860((119911 999492999484
119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously
we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have
119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we
can prove that (SVF8) holds
Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)
Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119910) Hence 119860 satisfies (SVF5)
Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38
Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF11) 119881119860(not119909 999492999484 119909) le 119881
119860(119909) for any 119909 isin 119871
(SVF12) 119881119860(sim 119909 rarr 119909) le 119881
119860(119909) for any 119909 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence
(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881
119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(not119909 999492999484 119909) le 119881
119860(119909)
Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40
The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem
Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881
119860(1) = 119881
119861(1) If 119860 is a
subpositive implicative vague filter of L so 119861 is
Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then
119909)) le 119881119860(119906 rarr 119909) Hence 119881
119860(119906 rarr 119909) = 119881
119860(1) = 119881
119861(1)
Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881
119860(119906 rarr 119909) = 119881
119861(1)
We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881
119861(119906) = imin119881
119861(119906 rarr
119909) 119881119861(119906) le 119881
119861(119909) Similarly we can show that 119866 satisfies
(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40
Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if
(BVF1) 119881119860(119909 or not119909) = 119881
119860(1) for all 119909 isin 119871
(BVF2) 119881119860(119909or sim 119909) = 119881
119860(1) for all 119909 isin 119871
Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters
Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860
(120572120573)is either empty or a Boolean filter of
119871 where 120572 le 120573
Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter
Proof Let119860 be a subpositive implicative vague filterWe have
119881119860 (1) = 119881
119860 (not (119909 or not119909) 999492999484 (119909 or not119909))
= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881
119860 (119909 or not119909)
(33)
Hence 119881119860(119909 or not119909) = 119881
119860(1) Similarly we can prove that 119860
satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881
119860(119909ornot119909) = 119881
119860(1)We have119881
119860(119909ornot119909) =
119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881
119860((119909 or not119909) 999492999484 119909) = imin119881
119860((119909 or not119909) 999492999484 119909) 119881
119860(119909 or not119909) le
119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence
119860 is a subpositive implicative vague filter byTheorem 41
Journal of Discrete Mathematics 9
6 Conclusion
In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters
Conflict of Interests
There is no conflict of interests regarding the publication ofthis paper
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009
[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009
[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972
[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001
[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007
[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends
in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998
[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011
[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010
[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006
an order le on elements 1198681= [1198861 1198871] and 119868
2= [1198862 1198872] of 119868[0 1]
asimin (119868
1 1198682) = [min 119886
1 1198862 min 119887
1 1198872]
1198681le 1198682
iff 1198861le 1198862 1198871le 1198872
(1)
Similarly we can define ge = and imax Then the concept ofimin and imax could be extended to define iinf and isup ofinfinite number of elements of 119868[0 1] It is a known fact that119871 = 119868[0 1] iinf isup le is a lattice with universal boundszero vague set and unit vague set For 120572 120573 isin [0 1] we nowdefine (120572 120573)-cut and 120572-cut of a vague set
Definition 11 (see [13]) Let 119860 be a vague set of a universe 119883
with the true-membership function 119905119860and false-membership
function119891119860The (120572 120573)-cut of the vague set119860 is a crisp subset
119860(120572120573)
of the set 119883 given by
119860(120572120573)
= 119909 isin 119883 119881119860 (119909) ge [120572 120573]
where 120572 le 120573
(2)
Clearly 119860(00)
= 119883 The (120572 120573)-cuts are also called vague-cutsof the vague set 119860
Definition 12 (see [13]) The 120572-cut of the vague set119860 is a crispsubset 119860
120572of the set119883 given by 119860
120572= 119860(120572120572)
Equivalently wecan define the 120572-cut as 119860
120572= 119909 isin 119883 119905
119860(119909) ge 120572
3 Vague Filters of Residuated Lattices
In this section we define the notion of vague filters ofresiduated lattices and obtain some related results
Definition 13 A vague set 119860 of a residuated lattice 119871 is calleda vague filter of 119871 if the following conditions hold
(VF1) if 119909 le 119910 then 119881119860(119909) le 119881
119860(119910) for all 119909 119910 isin 119871
(VF2) imin119881119860(119909) 119881119860(119910) le 119881
119860(119909lowast119910) for all 119909 119910 isin
119871
That is(1) 119909 le 119910 implies 119905
119860(119909) le 119905
119860(119910) and 1minus119891
119860(119909) le 1minus119891
119860(119910)
for all 119909 119910 isin 119871(2) min119905
119860(119909) 119905119860(119910) le 119905
119860(119909 lowast 119910) and min1 minus 119891
119860(119909) 1 minus
119891119860(119910) le 1 minus 119891
119860(119909 lowast 119910) for all 119909 119910 isin 119871
In the following proposition we will show that the vaguefilters of a residuated lattice exist
Proposition 14 Unit vague set and120572-vague set of a residuatedlattice 119871 are vague filters of 119871
Proof Let 119860 be a 120572-vague set of 119871 It is clear that 119860 satisfies(VF1) For 119909 119910 isin 119871 we have
119905119860(119909 lowast 119910) = 120572 = min 120572 120572 = min 119905
119860 (119909) 119905119860 (119910)
1 minus 119891119860(119909 lowast 119910)
= 120572 = min 120572 120572 = min 1 minus 119891119860 (119909) 1 minus 119891
119860(119910)
(3)
Thus imin119881119860(119909) 119881119860(119910) le 119881
119860(119909 lowast 119910) for all 119909 119910 isin 119871 Hence
it is a vague filter of 119871 The proof of the other cases is similar
Lemma 15 Let 119860 be a vague set of a residuated lattice 119871 suchthat it satisfies (VF1) Then the following are equivalent
(VF4) iminVA(x)VA(x rarr y) le VA(y)(VF5) iminVA(x)VA(x 999492999484 y) le VA(y)
for all 119909 119910 isin 119871
Proof Suppose that 119860 satisfies (VF4) We have 119909 le (119909 999492999484
119910) rarr 119910 By (VF1) 119881119860(119909) le 119881
119860((119909 999492999484 119910) rarr 119910) We get that
ge imin 119881119860 (119909) 119881119860 (119910)
(8)
Hence (VF2) holds
Theorem 17 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter if and only if it satisfies (VF5) and(VF6)
4 Journal of Discrete Mathematics
Definition 18 A vague set 119860 of a residuated lattice 119871 is calleda vague lattice filter of 119871 if it satisfies (VF1) and
(VLF2) imin119881119860(119909) 119881119860(119910) le 119881
119860(119909 and 119910) for all
119909 119910 isin 119871
Proposition 19 Let 119860 be a vague filter of a residuated lattice119871 then 119860 is a vague lattice filter
In the following example we will show that the converseof Proposition 19 may not be true in general
Example 20 Let 0 119886 119887 119888 1 such that 0 lt 119886 lt 119887 119888 lt 1where 119887 and 119888 are incomparable Define the operations lowast rarr and 999492999484 as follows
lowast 0 119886 119887 119888 1
0 0 0 0 0 0
119886 0 0 0 119886 119886
119887 0 119886 119887 119886 119887
119888 0 0 0 119888 119888
1 0 119886 119887 119888 1
997888rarr 0 119886 119887 119888 1
0 1 1 1 1 1
119886 119888 1 1 1 1
119887 119888 119888 1 119888 1
119888 0 119887 119887 1 1
1 0 119886 119887 119888 1
999492999484 0 119886 119887 119888 1
0 1 1 1 1 1
119886 119887 1 1 1 1
119887 0 119888 1 119888 1
119888 119887 119887 119887 1 1
1 0 119886 119887 119888 1
(9)
Then 119871 is a residuated lattice Let 119860 be the vague set definedas follows
In the following theorems we will study the relationshipbetween filters and vague filters of a residuated lattice
Theorem 21 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if for all 120572 120573 isin [0 1]the set 119860
(120572120573)is either empty or a filter of 119871 where 120572 le 120573
Proof Suppose that 119860 is a vague filter of 119871 and 120572 120573 isin [0 1]Let 119860
(120572120573)= 0 We will show that 119860
(120572120573)is a filter of 119871
(1) Suppose that 119909 le 119910 and 119909 isin 119860(120572120573)
By (VF1)119881119860(119910) ge
119881119860(119909) ge [120572 120573] Therefor 119910 isin 119860
(120572120573)
(2) Suppose that 119909 119910 isin 119860(120572120573)
Then 119881119860(119909) 119881119860(119910) ge
[120572 120573] By (VF2) we have 119881119860(119909 lowast 119910) ge
Hence (VF2) holdsSuppose that 119909 le 119910 119905
119860(119909) = 120572 and 1 minus 119891
119860(119909) = 120573 Hence
119909 isin 119860(120572120573)
By assumption 119860(120572120573)
is a filter We get that 119910 isin
119860(120572120573)
that is 119881119860(119910) ge [120572 120573] = [119905
119860(119909) 1 minus 119891
119860(119909)] = 119881
119860(119909)
Hence 119860 is a vague filter of 119871
The filters like119860(120572120573)
are also called vague-cuts filters of 119871
Corollary 22 Let 119860 be a vague filter of a residuated lattice 119871Then for all 120572 isin [0 1] the set 119860
120572is either empty or a filter of
119871
Corollary 23 Any filter 119865 of a residuated lattice 119871 is a vague-cut filter of some vague filter of 119871
Proof For all 119909 isin 119871 define
1 minus 119891119860 (119909) = 119905
119860 (119909) = 1 if 119909 isin 119865
120572 otherwise(13)
Hence we have 119881119860(119909) = [1 1] if 119909 isin 119865 and 119881
119860(119909) = [120572 120572]
otherwise where 120572 isin (0 1) It is clear that 119865 = 119860(11)
Let119909 119910 isin 119871 If 119909 119910 isin 119865 then 119909 lowast 119910 isin 119865 We have 119881
119860(119909 lowast 119910) =
[1 1] = imin119881119860(119909) 119881119860(119910) If one of 119909 or 119910 does not belong
to 119865 then one of119881119860(119909) and119881
119860(119910) is equal to [120572 120572]Therefore
119881119860(119909lowast119910) ge [120572 120572] = imin119881
119860(119909) 119881119860(119910) Suppose that 119909 le 119910
If 119909 isin 119865 then 119910 isin 119865 Hence 119881119860(119909) = 119881
119860(119910) If 119909 notin 119865 then
119881119860(119909) = [120572 120572] We have 119881
119860(119909) le 119881
119860(119910) Hence 119860 is a vague
filter of 119871
In the following proposition we will show that vaguefilters of a residuated lattice are a generalization of fuzzyfilters
Proposition 24 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if the fuzzy sets 119905
119860and
1 minus 119891119860are fuzzy filters in 119871
Journal of Discrete Mathematics 5
4 Lattice of Vague Filters
Let 119860 and 119861 be vague sets of a residuated lattice 119871 If 119881119860(119909) le
119881119861(119909) for all 119909 isin 119871 then we say 119861 containing119860 and denote it
by 119860 ⪯ 119861The unite vague set of a residuated lattice 119871 is a vague
filter of 119871 containing any vague filter of 119871 For a vague setwe can define the intersection of two vague sets 119860 and 119861 of aresiduated lattice 119871 by 119881
119860cap119861(119909) = imin119881
119860(119909) 119881119861(119909) for all
119909 isin 119871 It is easy to prove that the intersection of any family ofvague filters of 119871 is a vague filter of 119871 Hence we can define avague filter generated by a vague set as follows
Definition 25 Let 119860 be a vague set of a residuated lattice 119871A vague filter 119861 is called a vague filter generated by 119860 if itsatisfies
(i) 119860 ⪯ 119861(ii) 119860 ⪯ 119862 that implies 119861 ⪯ 119862 for all vague filters 119862 of 119871
The vague filter generated by 119860 is denoted by ⟨119860⟩
Example 26 Consider the vague lattice filter119860 in Example 20which is not a vague filter of 119871Then the vague filter generatedby 119860 is
⟨119860⟩ = ⟨0 [03 04]⟩ ⟨119886 [03 04]⟩
⟨119887 [03 04]⟩ ⟨119888 [03 08]⟩ ⟨0 [05 09]⟩
(14)
Theorem 27 Let 119860 be a vague set of a residuated lattice 119871Then the vague value of 119909 in ⟨119860⟩ is
119881⟨119860⟩ (119909) = isup imin VA (a1) VA (a2) VA (an)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
(15)
for all 119909 isin 119871
Proof First we will prove that ⟨119860⟩ is a vague set of 119871
119905⟨119860⟩ (119909)
= sup min 119905119860(1198861) 119905119860(1198862) 119905
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
le sup min 1 minus 119891119860(1198861) 1 minus 119891
119860(1198862) 1 minus 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= sup 1 minus max 119891119860(1198861) 119891119860(1198862) 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 1 minus inf max 119891119860(1198861) 119891119860(1198862) 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 1 minus 119891⟨119860⟩ (119909)
(16)
Now we will show that ⟨119860⟩ is a vague filter of 119871(VF1) Suppose that 119909 119910 isin 119871 is arbitrary For every 120576 gt 0
we can take 1198861 119886
119899 1198871 119887
119898isin 119871 such that119909 ge 119886
1lowastsdot sdot sdotlowast119886
119899
119910 ge 1198871lowast sdot sdot sdot lowast 119887
119898 and
119905⟨119860⟩ (119909) minus 120576 lt min 119905
119860(1198861) 119905119860(1198862) 119905
119860(119886119899)
119905⟨119860⟩
(119910) minus 120576 lt min 119905119860(1198871) 119905119860(1198872) 119905
119860(119887119898)
(17)
By Proposition 2 part (4) we have 119909 lowast 119910 ge 1198861lowast sdot sdot sdot lowast 119886
119899lowast 1198871lowast
sdot sdot sdot lowast 119887119898and then
119905⟨119860⟩
(119909 lowast 119910)
ge min 119905119860(1198861) 119905
119860(119886119899) 119905119860(1198871) 119905
119860(119887119898)
= min min 119905119860(1198861) 119905
119860(119886119899)
min 119905119860(1198871) 119905
119860(119887119898)
gt min (119905⟨119860⟩ (119909) minus 120576) (119905
⟨119860⟩(119910) minus 120576)
= min 119905⟨119860⟩ (119909) 119905⟨119860⟩ (119910) minus 120576
(18)
Since 120576 gt 0 is arbitrary we obtain that 119905⟨119860⟩
(119909 lowast 119910) ge
min119905⟨119860⟩
(119909) 119905⟨119860⟩
(119910)Similarly we can show that 1 minus 119891
⟨119860⟩(119909 lowast 119910) ge min1 minus
119891⟨119860⟩
(119909) 1 minus119891⟨119860⟩
(119910) Hence119881119860(119909lowast119910) ge imin119881
119860(119909) 119881119860(119910)
for all 119909 119910 isin 119871(VF2) Suppose that 119909 le 119910where 119909 119910 isin 119871 Let 119886
1 119886
119899isin
119871 such that 1198861lowast sdot sdot sdot lowast 119886
119899le 119909 Then 119886
1lowast sdot sdot sdot lowast 119886
119899le 119910 And we
get that
119881119860(119910) ge imin 119881
119860(1198861) 119881119860(1198862) 119881
119860(119886119899) (19)
Since 1198861 119886
119899isin 119871 where 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899is arbitrary then
119881⟨119860⟩
(119910) ge isup imin 119881119860(1198861) 119881119860(1198862) 119881
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 119881⟨119860⟩ (119909)
(20)
Therefore ⟨119860⟩ is a vague filter of 119871 by Theorem 16 It is clearthat119860 ⪯ ⟨119860⟩ Now let119862 be a vague filter of119871 such that119860 ⪯ 119862Then we have
119881⟨119860⟩ (119909)
= isup imin 119881119860(1198861) 119881
119860(119886119899) 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899
le isup imin 119881119862(1198861) 119881
119862(119886119899) 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899
= 119881119862 (119909)
(21)
for all 119909 isin 119871 Hence ⟨119860⟩ ⪯ 119862 and then ⟨119860⟩ is a vague filtergenerated by 119860
We denote the set of all vague filters of a residuated lattice119871 by VF(119871)
6 Journal of Discrete Mathematics
Corollary 28 (VF(119871) cap or) is a complete lattice
Proof Clearly if 119860119894119894isinΓ
is a family of vague filters of aresiduated lattice 119871 then the infimumof this family is⋂
119894isinΓ119860119894
and the supremum is ⋁119894isinΓ
119860119894= ⟨⋃119894isinΓ
119860119894⟩ It is easy to prove
that (IFF(119871) cap or) is a complete lattice
Let 119860 and 119861 be vague sets of a residuated lattice 119871 Then119860 lowast 119861 is defined as follows
119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910
ge imin 119881119860 (119911) 119881119861 (1) ge 119881
119860 (119911)
(26)
Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then
Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881
119862(119911) by (VF2)
Hence 119881119862(119911) ge isup119881
119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have
1 lowast 119911 = 119911 We obtain that
119881119862 (119911) = 119881
119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)
(2) It follows from Proposition 2 part (2) andDefinition 13
Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881
119860(1) = 119881
119861(1)
Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice
Journal of Discrete Mathematics 7
Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32
119860((119909 999492999484 119910) rarr 119910) for any
119909 119910 119911 isin 119871
Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies
119860((119909 rarr 119910) 999492999484 119910) ge
[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can
prove that 119860 satisfies (SVF4)
In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter
Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881
119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484
119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119910) 999492999484 119910)]) 119881119860(1) le 119881
119860((119909 rarr 119910) 999492999484 119910) Similarly we
can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter
Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
119860((119910 rarr 119911) 999492999484 119910) le 119881
119860((119910 rarr 119911) 999492999484
8 Journal of Discrete Mathematics
((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119911 999492999484 119910) Since 119860 is a vague
filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881
119860(119911 999492999484 119910) 119881
119860((119911 999492999484
119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously
we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have
119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we
can prove that (SVF8) holds
Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)
Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119910) Hence 119860 satisfies (SVF5)
Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38
Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF11) 119881119860(not119909 999492999484 119909) le 119881
119860(119909) for any 119909 isin 119871
(SVF12) 119881119860(sim 119909 rarr 119909) le 119881
119860(119909) for any 119909 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence
(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881
119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(not119909 999492999484 119909) le 119881
119860(119909)
Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40
The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem
Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881
119860(1) = 119881
119861(1) If 119860 is a
subpositive implicative vague filter of L so 119861 is
Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then
119909)) le 119881119860(119906 rarr 119909) Hence 119881
119860(119906 rarr 119909) = 119881
119860(1) = 119881
119861(1)
Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881
119860(119906 rarr 119909) = 119881
119861(1)
We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881
119861(119906) = imin119881
119861(119906 rarr
119909) 119881119861(119906) le 119881
119861(119909) Similarly we can show that 119866 satisfies
(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40
Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if
(BVF1) 119881119860(119909 or not119909) = 119881
119860(1) for all 119909 isin 119871
(BVF2) 119881119860(119909or sim 119909) = 119881
119860(1) for all 119909 isin 119871
Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters
Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860
(120572120573)is either empty or a Boolean filter of
119871 where 120572 le 120573
Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter
Proof Let119860 be a subpositive implicative vague filterWe have
119881119860 (1) = 119881
119860 (not (119909 or not119909) 999492999484 (119909 or not119909))
= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881
119860 (119909 or not119909)
(33)
Hence 119881119860(119909 or not119909) = 119881
119860(1) Similarly we can prove that 119860
satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881
119860(119909ornot119909) = 119881
119860(1)We have119881
119860(119909ornot119909) =
119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881
119860((119909 or not119909) 999492999484 119909) = imin119881
119860((119909 or not119909) 999492999484 119909) 119881
119860(119909 or not119909) le
119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence
119860 is a subpositive implicative vague filter byTheorem 41
Journal of Discrete Mathematics 9
6 Conclusion
In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters
Conflict of Interests
There is no conflict of interests regarding the publication ofthis paper
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009
[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009
[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972
[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001
[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007
[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends
in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998
[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011
[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010
[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006
Definition 18 A vague set 119860 of a residuated lattice 119871 is calleda vague lattice filter of 119871 if it satisfies (VF1) and
(VLF2) imin119881119860(119909) 119881119860(119910) le 119881
119860(119909 and 119910) for all
119909 119910 isin 119871
Proposition 19 Let 119860 be a vague filter of a residuated lattice119871 then 119860 is a vague lattice filter
In the following example we will show that the converseof Proposition 19 may not be true in general
Example 20 Let 0 119886 119887 119888 1 such that 0 lt 119886 lt 119887 119888 lt 1where 119887 and 119888 are incomparable Define the operations lowast rarr and 999492999484 as follows
lowast 0 119886 119887 119888 1
0 0 0 0 0 0
119886 0 0 0 119886 119886
119887 0 119886 119887 119886 119887
119888 0 0 0 119888 119888
1 0 119886 119887 119888 1
997888rarr 0 119886 119887 119888 1
0 1 1 1 1 1
119886 119888 1 1 1 1
119887 119888 119888 1 119888 1
119888 0 119887 119887 1 1
1 0 119886 119887 119888 1
999492999484 0 119886 119887 119888 1
0 1 1 1 1 1
119886 119887 1 1 1 1
119887 0 119888 1 119888 1
119888 119887 119887 119887 1 1
1 0 119886 119887 119888 1
(9)
Then 119871 is a residuated lattice Let 119860 be the vague set definedas follows
In the following theorems we will study the relationshipbetween filters and vague filters of a residuated lattice
Theorem 21 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if for all 120572 120573 isin [0 1]the set 119860
(120572120573)is either empty or a filter of 119871 where 120572 le 120573
Proof Suppose that 119860 is a vague filter of 119871 and 120572 120573 isin [0 1]Let 119860
(120572120573)= 0 We will show that 119860
(120572120573)is a filter of 119871
(1) Suppose that 119909 le 119910 and 119909 isin 119860(120572120573)
By (VF1)119881119860(119910) ge
119881119860(119909) ge [120572 120573] Therefor 119910 isin 119860
(120572120573)
(2) Suppose that 119909 119910 isin 119860(120572120573)
Then 119881119860(119909) 119881119860(119910) ge
[120572 120573] By (VF2) we have 119881119860(119909 lowast 119910) ge
Hence (VF2) holdsSuppose that 119909 le 119910 119905
119860(119909) = 120572 and 1 minus 119891
119860(119909) = 120573 Hence
119909 isin 119860(120572120573)
By assumption 119860(120572120573)
is a filter We get that 119910 isin
119860(120572120573)
that is 119881119860(119910) ge [120572 120573] = [119905
119860(119909) 1 minus 119891
119860(119909)] = 119881
119860(119909)
Hence 119860 is a vague filter of 119871
The filters like119860(120572120573)
are also called vague-cuts filters of 119871
Corollary 22 Let 119860 be a vague filter of a residuated lattice 119871Then for all 120572 isin [0 1] the set 119860
120572is either empty or a filter of
119871
Corollary 23 Any filter 119865 of a residuated lattice 119871 is a vague-cut filter of some vague filter of 119871
Proof For all 119909 isin 119871 define
1 minus 119891119860 (119909) = 119905
119860 (119909) = 1 if 119909 isin 119865
120572 otherwise(13)
Hence we have 119881119860(119909) = [1 1] if 119909 isin 119865 and 119881
119860(119909) = [120572 120572]
otherwise where 120572 isin (0 1) It is clear that 119865 = 119860(11)
Let119909 119910 isin 119871 If 119909 119910 isin 119865 then 119909 lowast 119910 isin 119865 We have 119881
119860(119909 lowast 119910) =
[1 1] = imin119881119860(119909) 119881119860(119910) If one of 119909 or 119910 does not belong
to 119865 then one of119881119860(119909) and119881
119860(119910) is equal to [120572 120572]Therefore
119881119860(119909lowast119910) ge [120572 120572] = imin119881
119860(119909) 119881119860(119910) Suppose that 119909 le 119910
If 119909 isin 119865 then 119910 isin 119865 Hence 119881119860(119909) = 119881
119860(119910) If 119909 notin 119865 then
119881119860(119909) = [120572 120572] We have 119881
119860(119909) le 119881
119860(119910) Hence 119860 is a vague
filter of 119871
In the following proposition we will show that vaguefilters of a residuated lattice are a generalization of fuzzyfilters
Proposition 24 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if the fuzzy sets 119905
119860and
1 minus 119891119860are fuzzy filters in 119871
Journal of Discrete Mathematics 5
4 Lattice of Vague Filters
Let 119860 and 119861 be vague sets of a residuated lattice 119871 If 119881119860(119909) le
119881119861(119909) for all 119909 isin 119871 then we say 119861 containing119860 and denote it
by 119860 ⪯ 119861The unite vague set of a residuated lattice 119871 is a vague
filter of 119871 containing any vague filter of 119871 For a vague setwe can define the intersection of two vague sets 119860 and 119861 of aresiduated lattice 119871 by 119881
119860cap119861(119909) = imin119881
119860(119909) 119881119861(119909) for all
119909 isin 119871 It is easy to prove that the intersection of any family ofvague filters of 119871 is a vague filter of 119871 Hence we can define avague filter generated by a vague set as follows
Definition 25 Let 119860 be a vague set of a residuated lattice 119871A vague filter 119861 is called a vague filter generated by 119860 if itsatisfies
(i) 119860 ⪯ 119861(ii) 119860 ⪯ 119862 that implies 119861 ⪯ 119862 for all vague filters 119862 of 119871
The vague filter generated by 119860 is denoted by ⟨119860⟩
Example 26 Consider the vague lattice filter119860 in Example 20which is not a vague filter of 119871Then the vague filter generatedby 119860 is
⟨119860⟩ = ⟨0 [03 04]⟩ ⟨119886 [03 04]⟩
⟨119887 [03 04]⟩ ⟨119888 [03 08]⟩ ⟨0 [05 09]⟩
(14)
Theorem 27 Let 119860 be a vague set of a residuated lattice 119871Then the vague value of 119909 in ⟨119860⟩ is
119881⟨119860⟩ (119909) = isup imin VA (a1) VA (a2) VA (an)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
(15)
for all 119909 isin 119871
Proof First we will prove that ⟨119860⟩ is a vague set of 119871
119905⟨119860⟩ (119909)
= sup min 119905119860(1198861) 119905119860(1198862) 119905
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
le sup min 1 minus 119891119860(1198861) 1 minus 119891
119860(1198862) 1 minus 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= sup 1 minus max 119891119860(1198861) 119891119860(1198862) 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 1 minus inf max 119891119860(1198861) 119891119860(1198862) 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 1 minus 119891⟨119860⟩ (119909)
(16)
Now we will show that ⟨119860⟩ is a vague filter of 119871(VF1) Suppose that 119909 119910 isin 119871 is arbitrary For every 120576 gt 0
we can take 1198861 119886
119899 1198871 119887
119898isin 119871 such that119909 ge 119886
1lowastsdot sdot sdotlowast119886
119899
119910 ge 1198871lowast sdot sdot sdot lowast 119887
119898 and
119905⟨119860⟩ (119909) minus 120576 lt min 119905
119860(1198861) 119905119860(1198862) 119905
119860(119886119899)
119905⟨119860⟩
(119910) minus 120576 lt min 119905119860(1198871) 119905119860(1198872) 119905
119860(119887119898)
(17)
By Proposition 2 part (4) we have 119909 lowast 119910 ge 1198861lowast sdot sdot sdot lowast 119886
119899lowast 1198871lowast
sdot sdot sdot lowast 119887119898and then
119905⟨119860⟩
(119909 lowast 119910)
ge min 119905119860(1198861) 119905
119860(119886119899) 119905119860(1198871) 119905
119860(119887119898)
= min min 119905119860(1198861) 119905
119860(119886119899)
min 119905119860(1198871) 119905
119860(119887119898)
gt min (119905⟨119860⟩ (119909) minus 120576) (119905
⟨119860⟩(119910) minus 120576)
= min 119905⟨119860⟩ (119909) 119905⟨119860⟩ (119910) minus 120576
(18)
Since 120576 gt 0 is arbitrary we obtain that 119905⟨119860⟩
(119909 lowast 119910) ge
min119905⟨119860⟩
(119909) 119905⟨119860⟩
(119910)Similarly we can show that 1 minus 119891
⟨119860⟩(119909 lowast 119910) ge min1 minus
119891⟨119860⟩
(119909) 1 minus119891⟨119860⟩
(119910) Hence119881119860(119909lowast119910) ge imin119881
119860(119909) 119881119860(119910)
for all 119909 119910 isin 119871(VF2) Suppose that 119909 le 119910where 119909 119910 isin 119871 Let 119886
1 119886
119899isin
119871 such that 1198861lowast sdot sdot sdot lowast 119886
119899le 119909 Then 119886
1lowast sdot sdot sdot lowast 119886
119899le 119910 And we
get that
119881119860(119910) ge imin 119881
119860(1198861) 119881119860(1198862) 119881
119860(119886119899) (19)
Since 1198861 119886
119899isin 119871 where 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899is arbitrary then
119881⟨119860⟩
(119910) ge isup imin 119881119860(1198861) 119881119860(1198862) 119881
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 119881⟨119860⟩ (119909)
(20)
Therefore ⟨119860⟩ is a vague filter of 119871 by Theorem 16 It is clearthat119860 ⪯ ⟨119860⟩ Now let119862 be a vague filter of119871 such that119860 ⪯ 119862Then we have
119881⟨119860⟩ (119909)
= isup imin 119881119860(1198861) 119881
119860(119886119899) 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899
le isup imin 119881119862(1198861) 119881
119862(119886119899) 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899
= 119881119862 (119909)
(21)
for all 119909 isin 119871 Hence ⟨119860⟩ ⪯ 119862 and then ⟨119860⟩ is a vague filtergenerated by 119860
We denote the set of all vague filters of a residuated lattice119871 by VF(119871)
6 Journal of Discrete Mathematics
Corollary 28 (VF(119871) cap or) is a complete lattice
Proof Clearly if 119860119894119894isinΓ
is a family of vague filters of aresiduated lattice 119871 then the infimumof this family is⋂
119894isinΓ119860119894
and the supremum is ⋁119894isinΓ
119860119894= ⟨⋃119894isinΓ
119860119894⟩ It is easy to prove
that (IFF(119871) cap or) is a complete lattice
Let 119860 and 119861 be vague sets of a residuated lattice 119871 Then119860 lowast 119861 is defined as follows
119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910
ge imin 119881119860 (119911) 119881119861 (1) ge 119881
119860 (119911)
(26)
Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then
Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881
119862(119911) by (VF2)
Hence 119881119862(119911) ge isup119881
119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have
1 lowast 119911 = 119911 We obtain that
119881119862 (119911) = 119881
119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)
(2) It follows from Proposition 2 part (2) andDefinition 13
Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881
119860(1) = 119881
119861(1)
Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice
Journal of Discrete Mathematics 7
Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32
119860((119909 999492999484 119910) rarr 119910) for any
119909 119910 119911 isin 119871
Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies
119860((119909 rarr 119910) 999492999484 119910) ge
[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can
prove that 119860 satisfies (SVF4)
In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter
Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881
119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484
119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119910) 999492999484 119910)]) 119881119860(1) le 119881
119860((119909 rarr 119910) 999492999484 119910) Similarly we
can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter
Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
119860((119910 rarr 119911) 999492999484 119910) le 119881
119860((119910 rarr 119911) 999492999484
8 Journal of Discrete Mathematics
((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119911 999492999484 119910) Since 119860 is a vague
filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881
119860(119911 999492999484 119910) 119881
119860((119911 999492999484
119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously
we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have
119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we
can prove that (SVF8) holds
Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)
Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119910) Hence 119860 satisfies (SVF5)
Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38
Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF11) 119881119860(not119909 999492999484 119909) le 119881
119860(119909) for any 119909 isin 119871
(SVF12) 119881119860(sim 119909 rarr 119909) le 119881
119860(119909) for any 119909 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence
(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881
119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(not119909 999492999484 119909) le 119881
119860(119909)
Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40
The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem
Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881
119860(1) = 119881
119861(1) If 119860 is a
subpositive implicative vague filter of L so 119861 is
Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then
119909)) le 119881119860(119906 rarr 119909) Hence 119881
119860(119906 rarr 119909) = 119881
119860(1) = 119881
119861(1)
Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881
119860(119906 rarr 119909) = 119881
119861(1)
We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881
119861(119906) = imin119881
119861(119906 rarr
119909) 119881119861(119906) le 119881
119861(119909) Similarly we can show that 119866 satisfies
(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40
Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if
(BVF1) 119881119860(119909 or not119909) = 119881
119860(1) for all 119909 isin 119871
(BVF2) 119881119860(119909or sim 119909) = 119881
119860(1) for all 119909 isin 119871
Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters
Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860
(120572120573)is either empty or a Boolean filter of
119871 where 120572 le 120573
Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter
Proof Let119860 be a subpositive implicative vague filterWe have
119881119860 (1) = 119881
119860 (not (119909 or not119909) 999492999484 (119909 or not119909))
= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881
119860 (119909 or not119909)
(33)
Hence 119881119860(119909 or not119909) = 119881
119860(1) Similarly we can prove that 119860
satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881
119860(119909ornot119909) = 119881
119860(1)We have119881
119860(119909ornot119909) =
119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881
119860((119909 or not119909) 999492999484 119909) = imin119881
119860((119909 or not119909) 999492999484 119909) 119881
119860(119909 or not119909) le
119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence
119860 is a subpositive implicative vague filter byTheorem 41
Journal of Discrete Mathematics 9
6 Conclusion
In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters
Conflict of Interests
There is no conflict of interests regarding the publication ofthis paper
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009
[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009
[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972
[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001
[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007
[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends
in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998
[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011
[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010
[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006
Let 119860 and 119861 be vague sets of a residuated lattice 119871 If 119881119860(119909) le
119881119861(119909) for all 119909 isin 119871 then we say 119861 containing119860 and denote it
by 119860 ⪯ 119861The unite vague set of a residuated lattice 119871 is a vague
filter of 119871 containing any vague filter of 119871 For a vague setwe can define the intersection of two vague sets 119860 and 119861 of aresiduated lattice 119871 by 119881
119860cap119861(119909) = imin119881
119860(119909) 119881119861(119909) for all
119909 isin 119871 It is easy to prove that the intersection of any family ofvague filters of 119871 is a vague filter of 119871 Hence we can define avague filter generated by a vague set as follows
Definition 25 Let 119860 be a vague set of a residuated lattice 119871A vague filter 119861 is called a vague filter generated by 119860 if itsatisfies
(i) 119860 ⪯ 119861(ii) 119860 ⪯ 119862 that implies 119861 ⪯ 119862 for all vague filters 119862 of 119871
The vague filter generated by 119860 is denoted by ⟨119860⟩
Example 26 Consider the vague lattice filter119860 in Example 20which is not a vague filter of 119871Then the vague filter generatedby 119860 is
⟨119860⟩ = ⟨0 [03 04]⟩ ⟨119886 [03 04]⟩
⟨119887 [03 04]⟩ ⟨119888 [03 08]⟩ ⟨0 [05 09]⟩
(14)
Theorem 27 Let 119860 be a vague set of a residuated lattice 119871Then the vague value of 119909 in ⟨119860⟩ is
119881⟨119860⟩ (119909) = isup imin VA (a1) VA (a2) VA (an)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
(15)
for all 119909 isin 119871
Proof First we will prove that ⟨119860⟩ is a vague set of 119871
119905⟨119860⟩ (119909)
= sup min 119905119860(1198861) 119905119860(1198862) 119905
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
le sup min 1 minus 119891119860(1198861) 1 minus 119891
119860(1198862) 1 minus 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= sup 1 minus max 119891119860(1198861) 119891119860(1198862) 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 1 minus inf max 119891119860(1198861) 119891119860(1198862) 119891
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 1 minus 119891⟨119860⟩ (119909)
(16)
Now we will show that ⟨119860⟩ is a vague filter of 119871(VF1) Suppose that 119909 119910 isin 119871 is arbitrary For every 120576 gt 0
we can take 1198861 119886
119899 1198871 119887
119898isin 119871 such that119909 ge 119886
1lowastsdot sdot sdotlowast119886
119899
119910 ge 1198871lowast sdot sdot sdot lowast 119887
119898 and
119905⟨119860⟩ (119909) minus 120576 lt min 119905
119860(1198861) 119905119860(1198862) 119905
119860(119886119899)
119905⟨119860⟩
(119910) minus 120576 lt min 119905119860(1198871) 119905119860(1198872) 119905
119860(119887119898)
(17)
By Proposition 2 part (4) we have 119909 lowast 119910 ge 1198861lowast sdot sdot sdot lowast 119886
119899lowast 1198871lowast
sdot sdot sdot lowast 119887119898and then
119905⟨119860⟩
(119909 lowast 119910)
ge min 119905119860(1198861) 119905
119860(119886119899) 119905119860(1198871) 119905
119860(119887119898)
= min min 119905119860(1198861) 119905
119860(119886119899)
min 119905119860(1198871) 119905
119860(119887119898)
gt min (119905⟨119860⟩ (119909) minus 120576) (119905
⟨119860⟩(119910) minus 120576)
= min 119905⟨119860⟩ (119909) 119905⟨119860⟩ (119910) minus 120576
(18)
Since 120576 gt 0 is arbitrary we obtain that 119905⟨119860⟩
(119909 lowast 119910) ge
min119905⟨119860⟩
(119909) 119905⟨119860⟩
(119910)Similarly we can show that 1 minus 119891
⟨119860⟩(119909 lowast 119910) ge min1 minus
119891⟨119860⟩
(119909) 1 minus119891⟨119860⟩
(119910) Hence119881119860(119909lowast119910) ge imin119881
119860(119909) 119881119860(119910)
for all 119909 119910 isin 119871(VF2) Suppose that 119909 le 119910where 119909 119910 isin 119871 Let 119886
1 119886
119899isin
119871 such that 1198861lowast sdot sdot sdot lowast 119886
119899le 119909 Then 119886
1lowast sdot sdot sdot lowast 119886
119899le 119910 And we
get that
119881119860(119910) ge imin 119881
119860(1198861) 119881119860(1198862) 119881
119860(119886119899) (19)
Since 1198861 119886
119899isin 119871 where 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899is arbitrary then
119881⟨119860⟩
(119910) ge isup imin 119881119860(1198861) 119881119860(1198862) 119881
119860(119886119899)
119909 ge 1198861lowast sdot sdot sdot lowast 119886
119899
= 119881⟨119860⟩ (119909)
(20)
Therefore ⟨119860⟩ is a vague filter of 119871 by Theorem 16 It is clearthat119860 ⪯ ⟨119860⟩ Now let119862 be a vague filter of119871 such that119860 ⪯ 119862Then we have
119881⟨119860⟩ (119909)
= isup imin 119881119860(1198861) 119881
119860(119886119899) 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899
le isup imin 119881119862(1198861) 119881
119862(119886119899) 119909 ge 119886
1lowast sdot sdot sdot lowast 119886
119899
= 119881119862 (119909)
(21)
for all 119909 isin 119871 Hence ⟨119860⟩ ⪯ 119862 and then ⟨119860⟩ is a vague filtergenerated by 119860
We denote the set of all vague filters of a residuated lattice119871 by VF(119871)
6 Journal of Discrete Mathematics
Corollary 28 (VF(119871) cap or) is a complete lattice
Proof Clearly if 119860119894119894isinΓ
is a family of vague filters of aresiduated lattice 119871 then the infimumof this family is⋂
119894isinΓ119860119894
and the supremum is ⋁119894isinΓ
119860119894= ⟨⋃119894isinΓ
119860119894⟩ It is easy to prove
that (IFF(119871) cap or) is a complete lattice
Let 119860 and 119861 be vague sets of a residuated lattice 119871 Then119860 lowast 119861 is defined as follows
119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910
ge imin 119881119860 (119911) 119881119861 (1) ge 119881
119860 (119911)
(26)
Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then
Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881
119862(119911) by (VF2)
Hence 119881119862(119911) ge isup119881
119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have
1 lowast 119911 = 119911 We obtain that
119881119862 (119911) = 119881
119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)
(2) It follows from Proposition 2 part (2) andDefinition 13
Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881
119860(1) = 119881
119861(1)
Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice
Journal of Discrete Mathematics 7
Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32
119860((119909 999492999484 119910) rarr 119910) for any
119909 119910 119911 isin 119871
Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies
119860((119909 rarr 119910) 999492999484 119910) ge
[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can
prove that 119860 satisfies (SVF4)
In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter
Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881
119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484
119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119910) 999492999484 119910)]) 119881119860(1) le 119881
119860((119909 rarr 119910) 999492999484 119910) Similarly we
can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter
Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
119860((119910 rarr 119911) 999492999484 119910) le 119881
119860((119910 rarr 119911) 999492999484
8 Journal of Discrete Mathematics
((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119911 999492999484 119910) Since 119860 is a vague
filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881
119860(119911 999492999484 119910) 119881
119860((119911 999492999484
119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously
we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have
119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we
can prove that (SVF8) holds
Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)
Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119910) Hence 119860 satisfies (SVF5)
Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38
Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF11) 119881119860(not119909 999492999484 119909) le 119881
119860(119909) for any 119909 isin 119871
(SVF12) 119881119860(sim 119909 rarr 119909) le 119881
119860(119909) for any 119909 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence
(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881
119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(not119909 999492999484 119909) le 119881
119860(119909)
Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40
The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem
Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881
119860(1) = 119881
119861(1) If 119860 is a
subpositive implicative vague filter of L so 119861 is
Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then
119909)) le 119881119860(119906 rarr 119909) Hence 119881
119860(119906 rarr 119909) = 119881
119860(1) = 119881
119861(1)
Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881
119860(119906 rarr 119909) = 119881
119861(1)
We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881
119861(119906) = imin119881
119861(119906 rarr
119909) 119881119861(119906) le 119881
119861(119909) Similarly we can show that 119866 satisfies
(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40
Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if
(BVF1) 119881119860(119909 or not119909) = 119881
119860(1) for all 119909 isin 119871
(BVF2) 119881119860(119909or sim 119909) = 119881
119860(1) for all 119909 isin 119871
Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters
Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860
(120572120573)is either empty or a Boolean filter of
119871 where 120572 le 120573
Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter
Proof Let119860 be a subpositive implicative vague filterWe have
119881119860 (1) = 119881
119860 (not (119909 or not119909) 999492999484 (119909 or not119909))
= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881
119860 (119909 or not119909)
(33)
Hence 119881119860(119909 or not119909) = 119881
119860(1) Similarly we can prove that 119860
satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881
119860(119909ornot119909) = 119881
119860(1)We have119881
119860(119909ornot119909) =
119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881
119860((119909 or not119909) 999492999484 119909) = imin119881
119860((119909 or not119909) 999492999484 119909) 119881
119860(119909 or not119909) le
119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence
119860 is a subpositive implicative vague filter byTheorem 41
Journal of Discrete Mathematics 9
6 Conclusion
In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters
Conflict of Interests
There is no conflict of interests regarding the publication ofthis paper
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009
[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009
[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972
[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001
[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007
[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends
in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998
[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011
[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010
[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006
119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910
ge imin 119881119860 (119911) 119881119861 (1) ge 119881
119860 (119911)
(26)
Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then
Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881
119862(119911) by (VF2)
Hence 119881119862(119911) ge isup119881
119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have
1 lowast 119911 = 119911 We obtain that
119881119862 (119911) = 119881
119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)
(2) It follows from Proposition 2 part (2) andDefinition 13
Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881
119860(1) = 119881
119861(1)
Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice
Journal of Discrete Mathematics 7
Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32
119860((119909 999492999484 119910) rarr 119910) for any
119909 119910 119911 isin 119871
Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies
119860((119909 rarr 119910) 999492999484 119910) ge
[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can
prove that 119860 satisfies (SVF4)
In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter
Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881
119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484
119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119910) 999492999484 119910)]) 119881119860(1) le 119881
119860((119909 rarr 119910) 999492999484 119910) Similarly we
can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter
Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
119860((119910 rarr 119911) 999492999484 119910) le 119881
119860((119910 rarr 119911) 999492999484
8 Journal of Discrete Mathematics
((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119911 999492999484 119910) Since 119860 is a vague
filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881
119860(119911 999492999484 119910) 119881
119860((119911 999492999484
119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously
we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have
119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we
can prove that (SVF8) holds
Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)
Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119910) Hence 119860 satisfies (SVF5)
Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38
Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF11) 119881119860(not119909 999492999484 119909) le 119881
119860(119909) for any 119909 isin 119871
(SVF12) 119881119860(sim 119909 rarr 119909) le 119881
119860(119909) for any 119909 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence
(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881
119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(not119909 999492999484 119909) le 119881
119860(119909)
Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40
The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem
Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881
119860(1) = 119881
119861(1) If 119860 is a
subpositive implicative vague filter of L so 119861 is
Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then
119909)) le 119881119860(119906 rarr 119909) Hence 119881
119860(119906 rarr 119909) = 119881
119860(1) = 119881
119861(1)
Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881
119860(119906 rarr 119909) = 119881
119861(1)
We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881
119861(119906) = imin119881
119861(119906 rarr
119909) 119881119861(119906) le 119881
119861(119909) Similarly we can show that 119866 satisfies
(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40
Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if
(BVF1) 119881119860(119909 or not119909) = 119881
119860(1) for all 119909 isin 119871
(BVF2) 119881119860(119909or sim 119909) = 119881
119860(1) for all 119909 isin 119871
Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters
Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860
(120572120573)is either empty or a Boolean filter of
119871 where 120572 le 120573
Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter
Proof Let119860 be a subpositive implicative vague filterWe have
119881119860 (1) = 119881
119860 (not (119909 or not119909) 999492999484 (119909 or not119909))
= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881
119860 (119909 or not119909)
(33)
Hence 119881119860(119909 or not119909) = 119881
119860(1) Similarly we can prove that 119860
satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881
119860(119909ornot119909) = 119881
119860(1)We have119881
119860(119909ornot119909) =
119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881
119860((119909 or not119909) 999492999484 119909) = imin119881
119860((119909 or not119909) 999492999484 119909) 119881
119860(119909 or not119909) le
119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence
119860 is a subpositive implicative vague filter byTheorem 41
Journal of Discrete Mathematics 9
6 Conclusion
In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters
Conflict of Interests
There is no conflict of interests regarding the publication ofthis paper
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009
[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009
[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972
[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001
[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007
[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends
in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998
[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011
[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010
[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006
Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32
119860((119909 999492999484 119910) rarr 119910) for any
119909 119910 119911 isin 119871
Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies
119860((119909 rarr 119910) 999492999484 119910) ge
[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can
prove that 119860 satisfies (SVF4)
In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter
Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le
VA(y) for any 119909 119910 119911 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881
119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484
119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119910) 999492999484 119910)]) 119881119860(1) le 119881
119860((119909 rarr 119910) 999492999484 119910) Similarly we
can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter
Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
119860((119910 rarr 119911) 999492999484 119910) le 119881
119860((119910 rarr 119911) 999492999484
8 Journal of Discrete Mathematics
((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119911 999492999484 119910) Since 119860 is a vague
filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881
119860(119911 999492999484 119910) 119881
119860((119911 999492999484
119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously
we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have
119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we
can prove that (SVF8) holds
Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)
Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119910) Hence 119860 satisfies (SVF5)
Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38
Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF11) 119881119860(not119909 999492999484 119909) le 119881
119860(119909) for any 119909 isin 119871
(SVF12) 119881119860(sim 119909 rarr 119909) le 119881
119860(119909) for any 119909 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence
(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881
119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(not119909 999492999484 119909) le 119881
119860(119909)
Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40
The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem
Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881
119860(1) = 119881
119861(1) If 119860 is a
subpositive implicative vague filter of L so 119861 is
Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then
119909)) le 119881119860(119906 rarr 119909) Hence 119881
119860(119906 rarr 119909) = 119881
119860(1) = 119881
119861(1)
Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881
119860(119906 rarr 119909) = 119881
119861(1)
We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881
119861(119906) = imin119881
119861(119906 rarr
119909) 119881119861(119906) le 119881
119861(119909) Similarly we can show that 119866 satisfies
(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40
Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if
(BVF1) 119881119860(119909 or not119909) = 119881
119860(1) for all 119909 isin 119871
(BVF2) 119881119860(119909or sim 119909) = 119881
119860(1) for all 119909 isin 119871
Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters
Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860
(120572120573)is either empty or a Boolean filter of
119871 where 120572 le 120573
Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter
Proof Let119860 be a subpositive implicative vague filterWe have
119881119860 (1) = 119881
119860 (not (119909 or not119909) 999492999484 (119909 or not119909))
= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881
119860 (119909 or not119909)
(33)
Hence 119881119860(119909 or not119909) = 119881
119860(1) Similarly we can prove that 119860
satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881
119860(119909ornot119909) = 119881
119860(1)We have119881
119860(119909ornot119909) =
119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881
119860((119909 or not119909) 999492999484 119909) = imin119881
119860((119909 or not119909) 999492999484 119909) 119881
119860(119909 or not119909) le
119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence
119860 is a subpositive implicative vague filter byTheorem 41
Journal of Discrete Mathematics 9
6 Conclusion
In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters
Conflict of Interests
There is no conflict of interests regarding the publication ofthis paper
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009
[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009
[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972
[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001
[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007
[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends
in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998
[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011
[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010
[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006
((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119911 999492999484 119910) Since 119860 is a vague
filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881
119860(119911 999492999484 119910) 119881
119860((119911 999492999484
119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously
we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have
119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we
can prove that (SVF8) holds
Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881
119860(119909) for any 119909 119910 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)
Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881
119881119860((119910 rarr 119911) 999492999484 119910) le 119881
119860(119910) Hence 119860 satisfies (SVF5)
Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38
Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies
(SVF11) 119881119860(not119909 999492999484 119909) le 119881
119860(119909) for any 119909 isin 119871
(SVF12) 119881119860(sim 119909 rarr 119909) le 119881
119860(119909) for any 119909 isin 119871
Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence
(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881
119860((119909 rarr 119910) 999492999484 119909) le 119881
119860(not119909 999492999484 119909) le 119881
119860(119909)
Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40
The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem
Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881
119860(1) = 119881
119861(1) If 119860 is a
subpositive implicative vague filter of L so 119861 is
Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then
119909)) le 119881119860(119906 rarr 119909) Hence 119881
119860(119906 rarr 119909) = 119881
119860(1) = 119881
119861(1)
Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881
119860(119906 rarr 119909) = 119881
119861(1)
We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881
119861(119906) = imin119881
119861(119906 rarr
119909) 119881119861(119906) le 119881
119861(119909) Similarly we can show that 119866 satisfies
(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40
Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if
(BVF1) 119881119860(119909 or not119909) = 119881
119860(1) for all 119909 isin 119871
(BVF2) 119881119860(119909or sim 119909) = 119881
119860(1) for all 119909 isin 119871
Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters
Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860
(120572120573)is either empty or a Boolean filter of
119871 where 120572 le 120573
Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter
Proof Let119860 be a subpositive implicative vague filterWe have
119881119860 (1) = 119881
119860 (not (119909 or not119909) 999492999484 (119909 or not119909))
= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881
119860 (119909 or not119909)
(33)
Hence 119881119860(119909 or not119909) = 119881
119860(1) Similarly we can prove that 119860
satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881
119860(119909ornot119909) = 119881
119860(1)We have119881
119860(119909ornot119909) =
119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881
119860((119909 or not119909) 999492999484 119909) = imin119881
119860((119909 or not119909) 999492999484 119909) 119881
119860(119909 or not119909) le
119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence
119860 is a subpositive implicative vague filter byTheorem 41
Journal of Discrete Mathematics 9
6 Conclusion
In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters
Conflict of Interests
There is no conflict of interests regarding the publication ofthis paper
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009
[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009
[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972
[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001
[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007
[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends
in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998
[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011
[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010
[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006
In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters
Conflict of Interests
There is no conflict of interests regarding the publication ofthis paper
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009
[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009
[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972
[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001
[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007
[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends
in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998
[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011
[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010
[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006