International Mathematical Forum, Vol. 7, 2012, no. 55, 2735 - 2742 Congruences in Hypersemilattices A. D. Lokhande Yashwantrao Chavan Warna College Warna nagar, Kolhapur, Maharashtra, India e-mail:[email protected]Aryani Gangadhara JSPM’s Rajarshi Shahu College of Engineering Tathawade, Pune, Maharashtra,India [email protected]Abstract In this paper we study congruence relation of hypersemilattices and Homomorphism and isomorphism of hypersemilattices using congruence relation and we prove that hyper meet of two congruence relations is a Congruence relation and is a fixed element of hypersemilattices. Also we prove embedding theorem for hypersemilattices using congruence relation. Finally we prove theorem on family of direct product of hypersemilattices Mathematics Subject Classification: 06B10, 06B99 Keywords: Hypersemilattices, Congruence relation, Homomorphism, Direct Product Introduction The Theory of hyperstructures was introduced in1934 by Marty [1] at the 8 th congress of Scandivinavian Mathematicians. This theory has been subsequently developed by the various authors. Some basic definitions and propositions about the hyperstructures are found in[3].Throughout this paper we are using definitions of
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⊗ (Cb1⊗Cb2)= (Cc1⊗Cd1) ⊗(Cc2⊗ Cd2) .Therefore Ψ is a congruence relation de fined on
L x K.
Definition 2.11 : If θ and Φ are congruence relations on L with θ ⊆ Φ then define
a relation Φ/ θ on L/ θ by (Cx , Cy ) ∈ Φ/ θ if and only if (x , y) ∈ Φ.
Theorem 2.12: Every congruence relation is the kernel of some homomorphism.
Proof: Let Φ is a congruence relation on L. Clearly Φ is a Equivalence relation on L/
θ, then to prove Φ/ θ is a congruence relation .Let (Cx , Cy ) and (Cz , Cw) ∈ Φ/ θ
( x, y) ,(z, w) ∈ Φ, This implies ( x ⊗ z) ,(y ⊗ w) ∈ Φ/ θ .That is (Cx ⊗z , Cy⊗ w)
Hence Φ/ θ is congruence relation on L/ θ. Now let Φ/ θ is a congruence relation on
L/ θ. Similarly we can prove Φ is a congruence relation on L.As Φ is a congruence
relation on L this implies (x ,y) ∈ Φ ⇔ (Cx , Cy) ∈ Φ/ θ ⇔ Cx = Cy ⇔ f(Cx) =f(Cy)
⇔ f(x ) =f(y) ⇔ Φ is kernel of homomorphism.
3. Hyper Meet of two congruence relations
Definition 3.1: Let A be the collection of all congruence relations defined on
hyperboolean algebra A. Then hyper meet of two congruence relations is denoted by
θ1 ⊗ θ2 and defined as Cx (θ1 ⊗ θ2) Cy
C x θ1 Cy and C x θ2 Cy.
Theorem3.2: Let θ1 and θ2 be any two congruence relations defined on an
hyperboolean algebra A .Define a relation θ1 ⊗ θ2 on A by C x (θ1 ⊗ θ2) C y ⇔ C x
θ1C y and C x θ2 Cy. Then θ1 ⊗ θ2 is a congruence relation defined on A such that θ1
⊗ θ2 is the fixed element of θ1 & θ2.
2740 A. D. Lokhande and Aryani Gangadhara
Proof: It is easy to prove that C x (θ1 ⊗ θ2) C x as C
x θ1 C x and C x θ2 C x. C x (θ1 ⊗
θ2) Cy y (θ1 ⊗ θ2) Cy as
C x θ1 Cy and C x θ2 Cy.
Cyθ1 C x and Cyθ2 C x
C x
θ1 Cy and C x θ2 Cy. now to prove transitivity, let C x (θ1 ⊗ θ2) Cy and Cy (θ1 ⊗ θ2) Cz,
by definition , C x θ1 Cy and C x θ2C y. Cy θ1 Cz and Cy θ2 Cz. This implies C x θ1 Cz and
C x θ2C z. Therefore C x (θ1 ⊗ θ2) Cz. Let f ∈F, let n be the corresponding integer, Let C xi (θ1 ⊗ θ2) Cyi for all i This implies C xi θ1 Cyi and C x i θ2 Cyi. That is f (C x 1, C x
2… C x n ) θ1f(Cy1, Cy2,…. Cyn ) and f(C x 1, C x 2,…. C x n ) θ2f(Cy1, Cy2,…. Cyn) that is
f (C x1, C x2,… C. xn) (θ1 ⊗ θ2) f(Cy1, Cy2,…. Cyn) . θ1 ⊗ θ2 is a hypercongruence
relation. To prove that θ1 ⊗ θ2 is fixed element in hypersemilattice L. θ1 ⊗ θ2 θ1and
θ1 ⊗ θ2 θ2 is obvious. That is [θ1 ⊗ θ2] ⊗ θ1 = {θ1 ⊗ θ2 } and [θ1 ⊗ θ2] ⊗ θ2 = {θ1 ⊗
θ2 }Therefore by [1.4], θ1 ⊗ θ2 is a fixed element of L.
Definition3.3 : Con (L)denotes the set of all congruence relations on a
hypersemilattice L. Then Con (L) forms complete Lattice with 0L and 1L, the fixed
(smallest) and (absorbent) Largest lement of congruence relations.
Theorem 3.4: For hypersemilattice L with 0L as fixed element and θ1, θ2 ∈ Con (L), then there is a natural embedding of L/ θ1 ⊗ θ2 L/ θ1 x L/ θ2.
Proof: Let Ψ = θ1 ⊗ θ2 .Then Φ / θ is a congruence relation on L/ θ1 ⊗ θ2 and let Φ
/ θ1,Φ / θ2 be congruence relations on L/ θ1 ,L/ θ2 respectively. Define f: L/ θ1 ⊗ θ2
L/ θ1 x L/ θ2 by f( (Cx) θ1 ⊗
θ2 ) ={( (Cx)
θ1 , (Cx)
θ2 ) / x ∈ L } .Define a
congruence relation Ψ by (Cx), Cy) ∈ Φ / θ if and only (x ,y) ∈ Φ. Let (Cx , Cy )
and (Cz Cw) ∈ Φ/ θ ( x, y) ,(z, w) ∈ Φ, This implies ( x ⊗ z) ,(y ⊗ w) ∈ Φ/ θ
.That is (Cx⊗z , Cy⊗ w) Hence Φ/ θ is congruence relation on L/ θ. Now let Φ/ θ is a
congruence relation on L/ θ. Similarly we can prove Φ/ θ1 and Φ/ θ2 are congruence
relations on L/ θ1 and L/ θ2. f( (C x) θ1 ⊗
θ2 ) = f( (Cy)
θ1 ⊗
θ2 )
( (C x)
θ1 , (C x)
θ2 )
=={( (Cy) θ1 , (Cy)
θ2 ) (C x)
θ1 = (Cy)
θ1 and (C x)
θ2 =(Cy)
θ2 (C x , Cy)
∈ θ1 and
(C x, Cy) ∈ θ2 (C x , Cy)
∈ θ1 ⊗ θ2 .But by Theorem [3.2],θ1 ⊗ θ2 is a fixed
element. That means by uniqueness property of fixed element of hypersemilattices θ1 ⊗ θ2 = 0L. Therefore C x = Cy. This implies f is one-one. To prove homomorphism
Let f( (C x ⊗ Cy ) θ1 ⊗
θ2) = [ (Cx ⊗ Cy )
θ1 , C x ⊗ Cy )
θ2 ] = ( C x , C x )
θ1 ⊗ θ2 ⊗ (Cy ,
Cy ) θ1 ⊗ θ2 = ( C x )
θ1 ⊗ θ2 ⊗ ( Cy )
θ1 ⊗ θ2.Hence f is homomorphism. L/ θ1 ⊗ θ2 L/ θ1
x L/ θ2.
Corollary 3.5: If hypersemilattice L has congruence relations θ1 & θ2 with θ1 ⊗ θ2=
0L then L L/ θ1 x L/ θ2 (an embedding).
Congruences in hypersemilattices 2741
Proof: Let Φ be a congruence relation on L and Φ/ θ1 , Φ/ θ2 be the congruence
relations on L/ θ1 and L/ θ2 respectively. It can be easily checked using theorem [3.4].
Then define Ψ : L L/ θ1 x L/ θ2 by Ψ( x Φ) = {( (C x) θ1 , (C x)
θ2 ) / x ∈ L } To prove
Ψ is one-one let Ψ(x Φ) = Ψ(y Φ) ( (C x)
θ1 , (C x)
θ2 ) =( (Cy)
θ1 , (Cy)
θ2 )
.This
implies ( (C x) θ1 = (Cy)
θ1 ) and
( (C x)
θ2 = (Cy)
θ2 ) , that is (x ,y) ∈ θ1 and (x ,y) ∈ θ2
therefore (x ,y) ∈ θ1 ⊗ θ2, but θ1 ⊗ θ2= 0L implies x=y. Let Ψ (x Φ ⊗ y Φ) = [(C x ⊗
Cy) θ1 , (C x ⊗ Cy)
θ2 ]= [(C x)
θ1 ⊗ (Cy)
θ1 ] ⊗ [(C x)
θ2 ⊗ (Cy)
θ2] = [(C x)
θ1 , (C x)
θ2
] ⊗ [(Cy) θ1 , (Cy)
θ2] = Ψ( x Φ) ⊗ Ψ( yΦ). Ψ is a homomorphism. Therefore A A/
θ1 x A/ θ2 is an embedding.
4. Direct Product of Hypersemilattices
Definition 4.1: Let (L, ⊗) and (S, ⊗ ) be Hypersemilattices. Define binary operation
on the Cartesian product Lx S as follows ( a1,b1) •(a2,b2) ={(c, d)/c ∈ a1 ⊗ a2,d ∈b1⊗
b2}for all ( a1,b1), (a2,b2) ∈ L x S ,then (L x S, •) is called the direct product of Hypersemilattices (L, ⊗ ) and (S, ⊗ ).
Definition 4.2: Let {Li/ i ∈ I} be a family of hypersemilattices. Then the direct
product of Li, i∈I is the Cartesian product (Li/i ∈I) = {(xi), i∈ I/xi ∈ Li}.
Theorem 4.3: The direct product of family of Hypersemilattices is again a hypersemilattice.
Proof: Let {Li/ i ∈ I} be a family of hypersemilattices. L = (Li/i ∈I) = {(xi), i∈ I/xi
∈ Li}. Define hyperoperation ⊗ on L as follows:
(xi) i ∈I ⊗ (yi) i ∈I ={(ti) i ∈I/ ti ∈ xi ⊗ yi). It is easy to observe that (xi) i ∈I ∈ (xi) i ∈I ⊗ (xi) i ∈I . (xi) i ∈I ⊗ (yi) i ∈I =(yi) i ∈I ⊗ (xi) i ∈I and for any xi , yi , zi ∈ Li , i ∈I, ((xi) i
∈I ⊗ (yi) i ∈I ) ⊗ (zi) i ∈I = {(pi) i ∈I pi∈ ti ⊗ zi } and (xi) i ∈I ⊗ ((yi) i ∈I ⊗
(zi) i ∈I) = {(ri) i ∈I ri∈ xi ⊗ qi}.Let (si) i ∈I ∈ (xi) i ∈I ⊗ ((yi) i ∈I ⊗ (zi) i ∈I).
Then (si) i ∈I ∈ {(pi) i ∈I /pi∈ ti ⊗ zi } for some ti∈ xi ⊗ yi si∈ ti ⊗ zi for
some ti∈ xi ⊗ yi si ∈ (xi ⊗ y) ⊗ zi
si ∈ xi ⊗ (y ⊗ zi ) si∈ xi ⊗ qi for
some qi∈ yi ⊗ zi (si) i ∈I ∈ {(ri) i ∈I ri∈ xi ⊗ qi} (si) i ∈I ∈ {(ri) i ∈I ri∈
xi ⊗ qi}=(xi) i ∈I ⊗ ((yi) i ∈I ⊗ (zi) i ∈I) and hence ((xi) i ∈I ⊗ (yi) i ∈I ) ⊗ (zi) i ∈I
(xi) i ∈I ⊗ ((yi) i ∈I ⊗ (zi) i ∈I).Similarly we can prove (xi) i ∈I ⊗ ((yi) i ∈I ⊗ (zi) i ∈I)
((xi) i ∈I ⊗ (yi)) i ∈I ) ⊗ (zi) i ∈I .Hence the theorem.
2742 A. D. Lokhande and Aryani Gangadhara
References
1. F.Marty, “Surune generalization de la notion de group” , 8
th Congress Math,
(1934)Pages 45-49 Scandinanes,Stockholm.
2. G.Gratzer “General Lattice theory”, 1998.
3. P.Corsini, Spaces J.Sets P.Sets, “Algebraic hyperstructures and applications”,
Hardonic Press, Inc (1994) P-45-53.
4. ZHAO Bin, XIAO Ying, HAN Sheng Wei,“Hypersemilattices”,