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Acta Math. Hungar., 2008 DOI: 10.1007/sl0474-007-7038-x
PROJECTIVELY CONDENSED SEMIGROUPS, GENERALIZED COMPLETELY
REGULAR
SEMIGROUPS AND PROJECTIVE ORTHOMONOIDS
Y. CHEN1 *, Y. HE2 * and K. P. SHUM3 *
1 School of Mathematical Sciences, South China Normal
University, Guangzhou, Guangdong 510631, China
e-mail: [email protected]
2 School of Computer Science, Hunan University of Science and
Technology, Xiangtan, Hunan 411201, China e-mail: [email protected]
3 Faculty of Science, The Chinese University of Hong Kong,
Shatin, N. T., Hong Kong e-mail: [email protected]
(Received February 22, 2007; revised July 25, 2007; accepted
August 6, 2007)
A b s t r a c t . T h e class PC of projectively condensed
semigroups is a quasivari-ety of una ry semigroups, t he class of
projective or thomonoids is a sub quasi variety of PC. Some
well-known classes of generalized completely regular semigroups
will be regarded as subquasivariet ies of PC. We give t he s t ruc
ture semilat t ice com-posit ion and t h e s t anda rd representa t
ion of projective or thomonoids , a n d t h e n obta in t he s t
ruc ture theorems of various generalized or thogroups .
1. Introduction
We follow the notations and conventions of Howie [13] and
Petrich and Reilly [19], especially for Green's equivalences on a
semigroup.
*Partially supported by the National Natural Science Foundation
of China (Grant No. 10771077) and the Natural Science Foundation of
Guangdong Province (Grant No. 021073; 06025062).
t Corresponding author. Partially supported by a grant of
Natural Scientific Foundation of Hu-nan (No. 06 J J2025) and a
grant of Scientific Research Foundation of Hunan Education
Department (No. 05A014).
^Partially supported by a UGC (HK) grant #2060123 (04-05). Key
words and phrases: P-condensed semigroup, quasivariety, generalized
completely regular
semigroup, P-orthomonoid, generalized orthogroup. 2000
Mathematics Subject Classification: 20M07, 08C15.
0236-5294/$ 20.00 © 2008 Akademiai Kiado, Budapest
mailto:[email protected]:[email protected]:[email protected]
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Y. CHEN, Y. HE and K. P. SHUM
Let S be a semigroup, and U a non-empty subset of the set E(S)
of all idempotents of S. For any a G S, the set of all [left,
right] idempotent identities of a is denoted by Ia [I
lai l£\, the intersection of U and Ia [I
la-, T£\
is denoted by Ua [Ua, U£\. The natural partial order ^ on E(S)
is a partial order relation defined as
= = {(e> / ) e E(S) x E(S) | / G Ie}.
If p is an equivalence on S such that |pa n U\ ^ 1 [especially,
|pa n Ua\ =1] for all a £ S, then 5 is said to be [strongly](p,
U)-surjective. In particular, if 5 is [strongly] (p,
E(S))-surjective, then S is said to be [strongly] p-surjective.
The right congruence C* and the equivalences £, Cu on 5 are
defined by
C* = { (a, b) G 5 x 5 | (Va;, y G S1) ax = ay
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PROJECTIVELY CONDENSED SEMIGROUPS
semigroups, the reader is referred to [3]." The classes in Table
1 with no reference number in the last column are introduced for
the first t ime in the
abundant
right abundant
left abundant
semiabundant
P-semiabundant
rpp
semi-rpp P-semi-rpp
lpp
semi-lpp
P-semidpp
£*, 7£*-surjective
£*, 7^-surjective
£, 7?*-surjective
C, 7^-surjective
(CU,U), (Ku,U)-smjective
£*-surjective
£-surjective
(Cu, t/)-surjective
72.*-surjective
7^-surjective
(1ZU, t/)-surjective
Ab
RAb
LAb
SeAb
PSeAb
Rpp
SeRpp
PSeRpp
Lpp
SeLpp
PSeLpp
[3]
[1] [14]
[2] [5]
[5]
[5]
Table 1
present paper. If S forms a P-semi-rpp semigroup with respect to
(the set of projections) U, then we write S(U) instead of writing
S. When U = E(S), we simplify the notation S(E(S)) to S. Similar
notations will be used without explanation.
PSeLpp f\
SeLpp
LAb
^ r PSeRpp SeRpp
Rpp
RAb
R^eAb^
Ab
Reg
Fig. 1
Recall tha t a semigroup S is regular if and only if it is
£-surjective (or alternatively, 7£-surjective). We denote the class
of regular semigroups by Reg. By virtue of Lemma 1.1 (i)-(iv), we
can see that the classes of semi-groups in Table 1 are
generalizations of the class Reg of regular semigroups.
Acta Mathematica Hungarica, 2008
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Y. CHEN, Y. HE and K. P. SHUM
In what follows, by a generalized regular semigroup we mean a
semigroup in the classes from Table 1, and then by a class of
generalized regular semi-groups we mean a class of semigroups from
Table 1. In fact, with respect to inclusion relation, Reg and the
classes of generalized regular semigroups form a semilattice with
the Hasse diagram shown in Fig. 1 above.
A semigroup S is called a completely regular semigroup if it is
Ti-surjective or, alternatively, each element of S is contained in
a subgroup of S. Since 7i = Cmionci semigroup, the class CReg of
completely regular semigroups is a subclass of Reg. The structure
of completely regular semigroups has been described in details by
Petrich and Reilly in [19]. During the recent decades, the
generalizations of completely regular semigroups in some classes of
gen-eralized regular semigroups have been investigated in a number
of papers (see for example [2]-[9], [14], [15], [20]). The aim of
this paper is to con-sider various generalizations of completely
regular semigroups in the classes of generalized regular semigroups
in a systematic way.
2. Projectively condensed semigroups
Recall from Petrich and Reilly [19] that a unary semigroup is a
triple (S, •, *) where (S, •) is a semigroup and the mapping * : a
*—>• a* is a unary operation on S. We will usually speak of a
unary semigroup (S, *) without mentioning the binary operation. If
(S, *) is a unary semigroup, then the set S* = {x* | x G S} is
called the set of projections of (S, *). A unary ho-momorphism
(especially, a unary isomorphism) of a unary semigroup (S, *) to a
unary semigroup (T, *) is a semigroup homomorphism (especially,
iso-morphism) compatible with the unary operation; a unary
congruence on a unary semigroup (S,*) is a congruence on the
semigroup S compatible with the unary operation. We denote the
quotient unary semigroup of a unary semigroup (S, *) modulo a unary
congruence p by (S, *)/p. If a non-empty subset T of a unary
semigroup (S, *) is closed under the operations on (S, *), then the
unary semigroup T with respect to the operations on (S, *) is
called a unary subsemtgroup of (S, *), and we denote it by (T,
*IT).
For two classes U and V of unary semigroups, we let U S V [U
< V] de-• roper ] Subc laSS Of V. A claSS U Of Unar* r
aPimicrrrama
•e exists a family X of implications (esp ;s of all semigroups
which satisfy each in
hi is defined by the family of implications X. we sometimes let
[(1), (2) , . . . , (n)] denote the quasivariety of unary semi-
note that li is a [proper] subclass of V. A class li of unary
semigroups is a iety if there exists a family X of implicatic it li
consists of all semigroups which satisfy
such a case, U is defined by the family of implications X. For
convenience,
quasivariety if there exists a family X of implications
(especially, identities) such that li consists of all semigroups
which satisfy each implication in X. In
i.se. is de.H.ned hit the fam.ilu of implications . For
couvenieni
groups defined by the sequence of implications marked by (1),
(2) , . . . , (n). Every quasivariety of unary semigroups is closed
under unary isomorphisms and unary subsemigroups (see [18, 19]). We
now list some implications as
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PROJECTIVELY CONDENSED SEMIGROUPS
below:
(1) x*x*=x*, x*xx*=x,
(2) y*xy*=x => y*x*y*=x*,
(3) XX = X => X* = X.
Denote the quasivarieties (1),(2) and its subquasivariety (1),
(2), (3) by VC and C, respectively. The following lemma is
evident.
LEMMA 2.1. Let (S,*) be a unary semigroup. Then the following
state-ments hold:
(i) (S, *) G [(1)] if and only if S* Q E(S) and a* G S* for all
a G S; (ii) (S, *) G [(2)] if and only if {x G S* \ xax = a} Q {x G
S* \ xa*x = x}
for all a G S; (iii) (S, *) eC if and only if (S, *) G VC and S*
= E(S).
Let S be a semigroup and U a non-empty subset of E(S). Define
the
equivalences Qu and Q on S as follows:
Qu = {(x,y) G S x S | Ux = Uy}, Q = {(x,y) G S x S \ Ix = Iy]
•
For any a G S, denote the Qu and Q-classes of S containing a by
Qua and
Qa, respectively. THEOREM 2.2. Let S be a semigroup. Then the
following statements are
equivalent: (i) S is endowed with a unary operation * such that
(S, *) G VC;
(ii) S is (Qu,U)-surjective for some non-empty subset U of E(S);
(iii) S is strongly (Qu,U)-surjective for some non-empty subset U
of E(S); (iv) there is a subset U of E(S) such that, for any a G S,
Ua has a mini-
mum element with respect to the natural partial order ^ on
E(S).
PROOF, (i) => (ii). Let (S, *) G VC and let U = S*. By Lemma
2.1(i) and
(ii), Ua = Ua* for all a G S, so a* G Q ̂ n U, and thus S is
(Qu, C7)-surjective.
(ii) => (iii). Suppose that S is {Qu, f/)-surjective. For any
a G S, if u, v
G Q^ n f/, then u G f/«, v (iv). Suppose that 5 is strongly
(Q^7, f/)-surjective. For any a e S,
denote the unique element of Q^ D U by a*. Then Ua* = Ua, and
hence a* ̂ v whenever v (i). Suppose that, for any a G S, the set
Ua has a minimum ele-ment a°u. Then we can routinely check that the
mapping °u : a ^ aOCJ is a well-defined unary operation on S such
that (5 , °u) G PC. •
vlcte Mathematica Hungarica, 2008
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Y. CHEN, Y. HE and K. P. SHUM
In the sequel, if it exists, the minimum element in Ua (a G S)
is denoted
by a°u. If S is (Qu, f/)-surjective, then denote the pair (S, U)
by S(U). The set of all such pairs is denoted by PC. We stipulate
that S(U) = T(V) in PC if and only if S = T and U = V. In case of U
= E(S), write a° and S instead of writing a°u and S[E(S)),
respectively, and denote the subset { S(U) G PC | U = E(S)} of PC
by C. The following result is useful.
THEOREM 2.3. The mappings r : VC -»• PC, (S, *) .-»• S(S*) and r
' : PC —> VC, T(V) i—> (T,°V) are mutually inverse
bijections, while the restriction
Q n T\C : C -»• C is a bijection of C onto C
PROOF. For any (S, *) e VC, it fi
S(S*) G PC which has property that
(2.1) (VaGS) a* = a°s*.
PROOF. For any (S, *) G VC, it follows by the proof of Theorem
2.2 that
Thus the mapping r is well-defined. If T(V) G PC, then, by the
proof of Theorem 2.2 again, (T, °V) G VC. Furthermore, since T°V g
V and v°V = v whenever v G V, we have
(2.2) T°V = V.
Therefore, the mapping r' is also well-defined. By using
equations (2.1) and (2.2), we can routinely check that
(S, *)TT' = S(S*)T' = (S, *), T(V)T'T = (T, °V)T = T(V).
Consequently, the mappings r and r ' are mutually inverse
bijections. By virtue of Lemma 2.1(iii), we claim that T\C is a
bijection from C onto C. •
In what follows, the members in VC and PC are called
protectively con-densed semigroups or simply V-condensed
semigroups; the members in C and C are called condensed semigroups.
For any subclass U of VC, denote the im-age { (S, *)T I (S, *) eU}
of U under the mapping r by U, and vice versa. We now give an
example of P-condensed semigroups.
EXAMPLE 2.4. Let S be an inverse semigroup with a complete
lattice of idempotents. Take a from S and denote the unique inverse
of a by a"1. It is evident that Ia is a non-empty subset of E(S)
since the identity Is of S lies in Ia. Suppose that e is the
maximum lower bound of Ia in E(S). For any / G Ia, since fa = a =
af, we have faa~
l = aa~l and a~laf = a~la. This implies that aa~l,a~la ^ / , so
that aa~l,a~la ^ e. Consequently, we have e G Ia and whence e = a°.
It follows that SeC. In particular, if S is either a finite inverse
monoid or the symmetric inverse semigroup on a set X, then S G
C.
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PROJECTIVELY CONDENSED SEMIGROUPS
3. General ized comple te ly regular semigroups
Analogous to Table 1, the following Table 2 gives the basic
information of some classes of semigroups:
superabundant H*-surjecti lve SuAb [3, 20]
semi-superabundant W-surjective SeSuAb [1] P-semi-superabundant
W^-surjective PSeSuA [4], [14]-[16]
super rpp (£* n ^ )-surjective SuRpp [9] strongly rpp strongly
£*-surjective StRpp [2, 6, 7]
strongly semi-rpp strongly £-surjective StSeRpp
strongly P-semi-rpp strongly (Cu, £/)-surjective StPSeRpp [11]
super lpp (£n^*)-surjective SuLpp
strongly lpp stron, gly 7£*-surjective StLpp
strongly semi-lpp strongly 72.-surjective StSeLpp
strongly P-semi-lpp strongly (Cu, £/)-surjective StPSeLpp
Table 2
In what follows, by a generalized completely regular semigroup
we mean a semigroup in the classes of semigroups from Table 2, and
by a class of generalized completely regular semigroups we mean a
class of semigroups from Table 2. In this section, we shall
consider the relationship between the classes of generalized
completely regular semigroups and the subquasivarieties of VC and C
partially satisfying the following implications:
(4)
(4)'
(5)
(4)'
xy* = x => x*y* = x*,
y*x = x => y*x* = x*,
xy = xz => x*y = x*z,
yx = zx => yx* = zx*.
The following theorem is one of the main results of this
paper.
THEOREM 3 .1 . Each class of generalized completely regular
semigroups is the image of a subquasivariety of VC under the
mapping r . Moreover, VC,
Cand such subquasivarieties are exactly all subquasivarieties of
VC partially satisfying the implications (3), (4), (4)', (5), (5)',
which form a semilattice with respect to inclusion relation. The
Hasse diagram of the above semilattice is shown in Fig. 2.
To establish Theorem 3.1, the following Lemma 3.2-Example 3.10
are needed.
Acta Mathematica Hungarica, 2008
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Y. CHEN, Y. HE and K. P. SHUM
PC = [(1),(2)]
[(l),(2),(3),^)']=StSeCPP
[(l),(2),(b)']=StCPP eSu
-f(l)>(3),(4)
SuAb = [(l),(5),(5)']
Fig. 2
StVSelZpp= [(1),(2),(4)]
StSeTZpp= [(1),(2),(3),(4)]
StTZPP=[(l),(2),(5)]
SuKPp=[(l),(4)',(5)]
LEMMA 3.2. [(1),(4),(4)'] S [(2)], [(1),(5)] S [(1),(4)],
[(1),(2),(5)]
S [(3)].
PROOF. Let (S, *) G [(1), (4), (4)']. Then S* g E(S). If x,y G S
such that y*xy* = x} then y*x = xy* = x. This implies by the
implications (4) and (4)' that y*x* = x*y* = x*, so y*x*y* = x*,
and hence (S,*) G [(2)] . This yields that [(1), (4), (4)'] ^
[(2)].
Let (S, *) G [(1), (5)] . If x,y e S such that xy* = x, then
x*x* = x* and xy* = xx*. This implies by the implication (5) that
x*y* = x*x* = x*, and hence (S, *) e [(4)] . This yields that [(1),
(5)] ^ [(1), (4)] .
Let (S, *) G [(1), (2), (5)]. Then (S, *) G VC. For any x G
E(S)J since x* G S*, we have x ^ x*. By the implication (5) and
Lemma 1.1 (v), we can see that x C* x*. This implies by Lemma
l.l(ii) that x C x*. Consequently, we have x = x*x = x*, and whence
(S, *) G [(3)] . Thus, [(1), (2), (5)]
^ [ ( 3 ) ] - • LEMMA 3.3. StPSeRpp = {S(U) G PC | Qu Q Cu} =
[(1), (2), (4)] r .
PROOF. Assume that S(U) G StPSeRpp. Take a from S and denote
the
unique element in L^ D Ua by a*. Let u be an arbitrary element
of Ua. Then u G Ura
(3.1)
= U^*, so that
a*u = a*.
This By using equation (3.1), we can routinely check that
(a*,ua*) G C\E,sy
implies by Lemma l.l(ii) that a Cua* CFua* CF{ua*)*. Since u,a*
G U a
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PROJECTIVELY CONDENSED SEMIGROUPS
such that ua* G E(S), we also have
(ua*)*a = (ua*)*ua*a = ua*a = a = aua* = aua*(ua*)* =
a(ua*),
thus (ua*)* G Lua n Ua, and so (ua*)* = a*. Furthermore, by
using equation (3.1) again, we get
(3.2) ua* = (ua*)*ua* = a*ua* = a*a* = a*.
Equations (3.1) and (3.2) illustrate that a* ^ u, and hence a* =
a°u. This implies by Theorem 2.2 that S(U) G PC. Moreover, if (a,
b) G Qu, then a*Qua Qub Qub*, thus a* = a°u = b°u = b*. This
implies that a Zua* = b*Cub, so that Qu g Cu. Now, we have
StPSeRpp Q { S(U) G PC | Qu Q Cu} .
Suppose that S(U) G PC on which Qu Q Cu. Then, by Theorem 2.2
and Theorem 2.3, we can see that (S, °u)= S(U)T' G VC with S°u = U.
Further-more, since (a, a°u) G Qu g Zu whenever a G S, we claim
that (S, °u) G [(4)] also. Consequently, we get
{S(U) G PC | Qu Q Cu} Q [(1), (2), (4)] r .
Let (S, *) G [(1), (2), (4)] and let U = S*. For any a G S,
since a* = a°u', we have Ura* Q IFa. Furthermore, by implication
(4), we can see that IFa Q U
ra*
also. Therefore, we have a* G L^ n Ua. Also, if u G L^ n Ua,
then u Cu a°u
^ u. By Lemma 1.1(h), we have u C a°u ^ u, so that a°u = ua°u =
u. This
implies that a°u is the unique element in Lua n Ua. Thus S(U) =
(S, *)r G StPSeRpp. Now, we have
[(1), (2), (4)] r i StPSeRpp. •
COROLLARY 3.4. StSeRpp = {S G C | Q Q £} = [(1), (2), (3), (4)]
r .
PROOF. This result follows from Lemma 3.3 and Lemma 2.1 (hi).
•
COROLLARY 3.5.
StRpp = StSeRpp n Rpp = {S G C | Q Q £*} = [(1), (2), (5)]
r.
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10 Y. CHEN, Y. HE and K. P. SHUM
PROOF. By Lemma 1.1 (iii) and Corollary 3.4, we see that
StRpp = StSeRpp n Rpp = {S e C | Q g £*}.
If S G C on which Q g £*, then (S, °) = SreC such that (a, a°) G
Q gC* for all aeS. By Lemma l.l(v), we have (S,°) G [(5)], whence
(S,°) e [(1),(2),(3),(5)]. Furthermore, by Lemma 3.2, we claim that
[(1), (2), (3), (5)] = [(1), (2), (5)] . Thus {S G C | Q g £*} g
[(1), (2), (5)] r .
Conversely, if (S,*) G [(1), (2), (5)] , then (S,*) eCn [(5)]
follows by Lemma 3.2. Furthermore, for any a G S, by the
implication (5) and Lemma l.l(v), we can see that (a,a*) G £*. This
implies that Q g £*, and so that (S,*)T G C on which Q g C*. Thus
[(1), (2), (5)] r Q {S G C | Q Q £*} also. D
LEMMA 3.6.
PSeSuAb = {S(U) G PC | Qu = Hu} = StPSeRpp n StPSeLpp
= [(1),(4), (4)']r.
PROOF. Let S be a semigroup, and U a non-empty subset of E(S).
For
any (a, 6) G Hu, since 7^^ = Cun7^^, we have Ua = UliC\Ula =
UlC\U
lb = [/&,
whence (a, 6) G Q^7. Therefore, Hu g Q^.
It is evident that (
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PROJECTIVELY CONDENSED SEMIGROUPS 11
PROOF. This result follows from Corollary 3.6, Corollary 3.4 and
its dual.
• COROLLARY 3.8. SuRpp = {S G C | Q = C* n 1Z} = StRpp n StSeLpp
=
PROOF. By Lemma l.l(iii) and Corollary 3.7,
SuRpp = Rpp n SeSuAb = Rpp n StSeRpp n StSeLpp = StRpp n
StSeLpp.
By Lemma 3.5 and the dual result of Corollary 3.4, we have
StRpp n StSeLpp = {S G C | Q Q £*} n {S G C | Q Q K}
= {s e C I Q
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12 Y. CHEN, Y. HE and K. P. SHUM
Then S is a finite inverse monoid. In fact, S is isomorphic to
B^, the five ele-ment Brandt semigroup with an identity adjoined.
This implies by Example 2.4 that (S, °) G C, where
a° = 6° =1° = 1 , e° = e, / ° = / , 0° = 0 .
Since af = a but 1/ = / = 1 , we claim that (5,°) £ [(4)] . This
example demonstrates that
StVSeKpp < VC, StSeKpp 0 and n ^ 0. Then, 5/£* = { Au {1},
£>} and S/TZ* = {Au B, {1}} . It follows that S G StRpp, so that
(S, °) G Siftpp where 1° = ( a m ) ° = 1 , (6ra)° = e. Since 6a =
aa but 61= b = a = a1, we claim that (S, °) ^ [(5)\ . 1 his example
demonstrates that
VSeSuAb < StVSeTZpp, SeSuAb < StSeRpp, SuAb <
StTlpp.
(iv) If S is a left cancellative monoid but not right
cancellative, then S is a unipotent semigroup (i.e., a monoid with
a unique idempotent); furthermore, we can easily show that S G
SuRpp but S $. SuAb. Thus SuAb< SuTZpp.
(v) If S is a monoid which is neither left cancellative nor
right cancella-tive, then S G SeSuAb but S $. SuRpp. Thus SuTZpp
< SeSuAb.
P R O O F OF THEOREM 3.1. By Lemma 3.3-Corollary 3.9 and their
dual results, we can obtain the equations in Fig. 2. Example 3.10
shows that the quasivarieties of unary semigroups in Fig. 2 are
pairwise distinct. By using Lemma 3.2 and its dual result, we can
routinely check that the subquasi-varieties of VC in Fig. 2 are
exactly the subquasivarieties of VC partially satisfying the
implications (3), (4), (4)', (5), (5)', which form a lower lattice
with the given Hasse diagram. We omit the details. •
COROLLARY 3.11. A semigroup S is in CReg if and only if it is
strongly C-surjective.
PROOF. If S G CReg, then, by Lemma 1.1 (iv) and Corollary 3.9, S
is strongly £-surjective. Conversely, suppose that S is a strongly
£-surjective semigroup. Then, of course, S is a regular semigroup.
Moreover, for any a G S, the unique element in La n Ia is the
minimum idempotent identity a° of a. Take a+ G V(a) such that a+a =
a°. Then, by direct computation, a) = a°a+a° G V(a) and a° G Ra\ n
/ a t - Clearly, (a))° G La\ V\Ia\. Since (a})°
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PROJECTIVELY CONDENSED SEMIGROUPS 13
is the minimum idempotent identity of a\ we have (af)° ^ a°.
Suppose that a' G V(a)) such that a)a' = a° and a'a) =(a))°. Then
we have
(at)° = (at)°a° = (a t )Va ' = a^a' = a°.
This implies that aa^ TZ a C a° C af C aa"1, and hence aa^TL a.
Thus S G CReg also. D
EXAMPLE 3.12. By Corollary 3.11, Lemma 1.1 and Lemma 3.6, the
classes of generalized completely regular semigroups are
generalizations of CReg. Unfortunately, CTZeg is not a
subquasivariety of VC since CTZeg is not closed under taking unary
subsemigroups. For example, let Z be the addi-tive group of
integers and N be the additive semigroup of natural numbers. For
any n G Z, we have n° =0. Then (Z, °) G CTZeg and (iV, °\N) is a
unary subsemigroup of (Z,°). Clearly, (N,°\N) £ CTZeg.
4. Projective orthomonoids
In the literature, a completely regular semigroup S is called an
orthogroup if E(S) forms a subsemigroup. Of course, the class of
orthogroups OG is a proper subclass of CReg. Since orthogroups are
exactly the semilattices of rectangular groups (see [19]), a
completely regular semigroup S is an or-thogroup if and only if the
unary operation ° on S satisfies the identities
{x°y0)° = x°y°, (xy)0x°y0(xy)0 = (xy)°, x°y0(xy)0x°y0 =
x°y°.
If (S, *) G VC satisfies the identities
(6) (x*y*)*=x*y*, (xy)* x* y* (xy)* = (xy)*, x* y* (xy)* x* y* =
x* y*,
then we call (S, *) a projective orthomonoid or simply a
V-orthomonoid. We denote the quasivariety of P-orthomonoids by VOM.
In what follows, if it is necessary, we write a relation p on a
semigroup S as p(S).
LEMMA 4.1. Let (S, *) G VC. Then (S, *) G VOM if and only if S*
is a subsemigroup of S and (xy)* V(S*) x*y* for all x,y G S.
PROOF. By Theorem 2.2, (S, *) satisfies the identity (x*)* = x\
and thus (S,*) satisfies the identity (x*y*)* = x*y* if and only if
S* is a subband of S. If this is the case, the elements x and y of
S satisfy the identities (xy)*x*y*(xy)* = (xy)* and x*y*(xy)*x*y* =
x*y* if and only if (xy)* and x*y* are mutually inverse, and if and
only if (xy)* V(S*) x*y*. •
Acta Mathematica Hungarica, 2008
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14 Y. CHEN, Y. HE and K. P. SHUM
EXAMPLE 4.2. (i) If S is the direct product of a monoid T and a
rectan-lar band IxA, then we write S as / x T x A. In this case,
the mapping (i,t, A) i—> (i, IT, A) is a unary operation on S
such that (S, *) G VOM.
una.rv semipronn that is una.rv isomornhic to such a.
-orthomonoid is
(ii) Clearly, CVOM = VOM n [x*y = yx*] is the quasivariety of
P-ortho central projections. If T is a semilattice Y of monok
called a V-plank. l̂ee
monoids with central projections. If T is a semilattice Y of
monoids Ta (a G Y) and E = {lTa | a G 7 } is a subsemigroup of T,
then T is a strong semilattice Y of the monoids Ta (a G Y) and -B
lies in the center of T (see [4]). Moreover, we can easily see that
T(E) G CPOM. We call the P-condensed semigroup T{E) an
E-semilattice of monoids.
Let (S, *) G VC. We define the following equivalences on the
semigroup S:
L = { (a,6) G S x S | a* C b*} , TZ = { (a,b) G 5 x 5 | a* 72-
6*} , f> = £\JTZ.
For any a E S, the £, 7£ and P-classes of a in 5 are denoted by
L a, R a and D a, respectively. Then, it is evident that CnTZ =
0
s*. We use S =[Y; Sa] to denote that a semigroup S is a
semilattice Y of the
semigroups Sa (a G Y). Assume that S =[Y; Sa] and each Sa is
equipped with a unary operation *a. For any x G S, we let a;* =
a;*" when x £ Sa. Then the mapping * : a; i—> a?* is a unary
operation on 5. We call the unary semigroup (S,*) a unary
semilattice of (Sa,*
a)(a G Y), in notation (S,*) = [Y;(Sa,*
a)] • We call the unary semigroup (Y, *) formed by defining a
unary operation
* : x i—> a; on a semilattice Y a unary semilattice. If p is
a unary congru-ence on a unary semigroup (S, *) such that (S, *)/ p
is a unary semilattice, then p is called a unary semilattice
congruence on (S, *). If this is the case, p is a semilattice
congruence on S. Let p be a unary semilattice congruence on a unary
semigroup (S, *), and let the semilattice decomposition of S
in-duced by p be [Y; Sa]. Then, for any x G Sa (a G Y), we can
easily see that a ^ = (a^ )* = x*p^ in (Y, * )= (S , *)/p. This
implies that x* G 5 a also, and thus the mapping *'s« : x —>
a?*'s« = a;* is a unary operation on 5 a such that (S, * )= [Y ; (S
a , *lsa)] . We call [Y ; (S a , * ^ ) ] ^ c unary semilattice
decom-position of (S, *) induced by p.
LEMMA 4.3. Let (S, *) G POM. Then the following statements hold:
(i) V = {(a, b) G 5 x 5 | a* £>(5*) 6*} = CoTZ = 1Zo C;
(ii) P is i/ie minimum unary semilattice congruence on (S, *);
(iii) »/ L> is a P-c/ass o/ 5, t/ien {D, *b) is 0 V-plank.
PROOF, (i) Let /C = { (a, 6) G 5 x S \ a* T>(S*) b*} . Since
S* is a subband
of 5, we have C(S)\ c , = C(S*) and K(S)\ Qt = K(S*). This
implies that
vlcte Mathematica Hungarica, 2008
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PROJECTIVELY CONDENSED SEMIGROUPS 15
£,KglC, and so that IoK Q V Q /C. For any (c,d) G /C, since c*
£>(S*) d*, there exists u = TZo £ also.
(ii) By virtue of the statement (i), V\s* = V(S*) and
(4.1) (VaeS ) ( a , a * ) e P .
For any (a, 6), (c, d) G £>, by the formula (4.1), we have
(a*, 6*), (c*,d*) G £>|,s* = 2?(5*). Furthermore, since the
Green's equivalence V(S*) on the band 5* is the minimum semilattice
congruence, we also have (a*c*,b*d*) G V(S*). This implies by Lemma
4.1 that
(ac)* V(S*) a*c* V(S*) b*d* V(S*) (bd)*,
and so that (ac, bd) G V. Now, we have proved that V is a
congruence on the semigroup S. The formula (4.1) also illustrates
that (a, b) G V always implies (a*,b*) et>. Thus V is a unary
congruence on (S, *). Once again, by using the formula (4.1), we
can routinely show that (S, *)/f> is a unary semilattice.
Thereby, T> is a unary semilattice congruence on (S, *).
Assume that 5 is also a unary semilattice congruence on (S, *).
Then (Sis* is a semilattice congruence on S*, so that £>|,s* =
T>(S*) Q 5\st. For any (a,b) G £>, since (a*,b*) G P b * . we
have (a*,b*) G #1 OHl, whence a5^ = a*^ _ 5*^ _ 5 ^ ^his implies
that V Q 5. Thus V is the minimum unary semilattice congruence on
(S, *).
(iii) Let D be a P-class in S. Then, by the statement (ii), we
can see that (D, *b) is a unary subsemigroup of (S, *), and thus
(15, *b) e POM. Furthermore, since V\s* = V(S*), the set D*\D of
projections of (D, *b) is a maximal rectangular subband of S*.
Choose u from _D*b and let Du = {x G D | a;*b = «}. For any a, 6
G -DM,
it is evident that u G D^bD (i.e., « is an identity of ab in
D*b). Since (a6)*'D
is the minimum identity of ab in £)*b and the elements of the
rectangu-lar band £>*b are primitive, we have u = (a&)*b,
whence a& G £>„• This implies that Du is a subsemigroup of
D. Clearly, u is the identity of Du. Let T be the direct product Du
x _D*b of the monoid Du and the rect-angular band £>*b. Then T
forms a P-plank with respect to the unary operation * : (t,d)
i—> (u,d). We can routinely show that the mapping a : x i—>
(TO-U, a;*b) is a unary isomorphism from (£), *b) onto (T, *).
D
In what follows, we call the semilattice decomposition of a
P-orthomonoid (S, *) induced by the relation V the structure
semilattice decomposition of
Acta Mathematica Hungarica, 2008
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16 Y. CHEN, Y. HE and K. P. SHUM
(S,*). By the notation (S,*)= [Y;(Sa,*")] G VOM, we always
consider that (S, *) G VOM and [Y;(Sa, *
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PROJECTIVELY CONDENSED SEMIGROUPS 17
Standard representation of semigroups is an important tool for
construct-ing bands, completely regular semigroups, orthodox super
rpp semigroups, superabundant semigroups and orthodox semigroups
(see [18, 19, 9, 20, 10]). We now construct P-orthomonoids by using
this method.
Let (S, *) G POM and define the relation 7 on (S, *) as
follows:
7 = { (a, b) G S x S I a*ba* = a, b*ab* = b} .
We first give some characterizations of the relations 7, C and
V, as well as their relationships.
LEMMA 4.6. Let (S, * )= [Y;(Sa, *
-
18 Y. CHEN, Y. HE and K. P. SHUM
dm L d*m. Since (c*,d*) G C and C is a right congruence on S, by
Lemma l.l(ii) and the above statements, we claim that (c*m,d*m) G C
Q Cu = C. Thereby, L is a right congruence on S.
(vi) It follows from the statement (ii) that 7 is an equivalence
on S and, for any a G Sa,
(4.3) a7 = Ia x {Sa} x A a .
Choose (a, b) from 7 and c from S. Then, by (4.3), a,b G Sa and
c G Sp for
some a,{3&Y. Since a/3 ^ a and 5 * ^ is a maximal
rectangular subband of
S*, we have -u6*v = uv for all u, v G ̂ J^f, whence
(ac)*6c(ac)* = (ac)*b* ab* c(ac)* = (ac)* b* (ab* c)* ab*
c(ac)*
= (ac)* (ab* c)* ab* c(ac)* = (ac)* ab* c(ac)*
= (ac)*a( (ac)*a) *b*(c(ac)*) *c(ac)*
= (ac)*a((ac)*a)*(c(ac)*)*c(ac)* = (ac)*ac(ac)* = ac.
By using similar arguments as above, we can also show that
(bc)*ac(bc)* = be. Thus 7 is a right congruence on S. The dual is
that 7 is a left congruence on S. Furthermore, by using (4.3) once
again, we derive that a* =(Fa , lTa,Ta) 7 (Fb, lTa,Tb)= b*, whence
(a*,b*) G 7. Thus, 7 is a unary congruence on (S,*).
Let (S, *)/7 = (T, +). For any a G Y, let Ta = {arf \ a G Sa}
and define a unary operation +" : a^ 1—> a*^ on Ta. It is a
routine matter to show that (T, + ) = [Y;(Ta, +«)] , where each Ta
is a monoid isomorphic to Ta, and the unary operation +a on Ta can
be alternatively defined by
+a : x 1—>• lya . Fur-thermore, since i£ = {lya | a G F } =
S*7
1', we claim that _B is a subsemigroup of f. This implies by
Remark 4.5 that (T, + ) / 7 G CPOM.
If 5 is also a unary congruence on (S, *) such that (S, *)/5 G
CVOM, then 5|s* is a semilattice congruence on S*, and so ^\s*
^S\s*. Consequently, for any (a, b) G 7, we have (a*,b*) G 5, and
hence 6 = b*ab* 5 a*aa* = a. There-fore jg5. •
Denote the set of transformations on a non-empty set X by T(X).
If £ G T(X) is a constant on X, then denote £ and the value of £ by
[£]. For any i G X, denote the constant on X with value i by [i].
The semigroup with underlying set T(X) under the composition 7̂7 :
a; 1—̂ (x£)r) is denoted by T(X) also. The dual semigroup of T(X)
is denoted by T*(X), whose element are assumed to act X on the left
side.
Acta Mathematica Hungarica, 2008
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PROJECTIVELY CONDENSED SEMIGROUPS 19
THEOREM 4.7. Let (T, + ) = [Y;(Ta, +«)] e CVOM. For any aeY, let
Ia and Aa be non-empty sets such that Ia n Ip = Aa n A/? = whenever
a / j3 in Y. Form the set Sa = Ia x Ta x Aa for any aeY, and let S
= UQ,eySQ,. Assume that, for any n ^ a in Y, there exist
mappings
£a,n '• Sa —> T*(IK), a i—> ^ K and r\a%K : Sa —>
T(AK), ai-»^K,
which satisfy the following conditions: for any (i,x,\) e S a
and (j,y,/J>)
£ Sp,
(C.l) Cj'«'A) = [i\, r?J'«'A) = [A];
(C.2) (3(k,v)eIaf}xAaf}) &*f*^f = [k], V{ZfvfVaf = [vY,
(C.3) (VK ^ a/3) ^ y , I / ) = Cj'«'A) * ̂ , / i ) , r ? ! ^ ' ^
= rfc'^nf™)-
Define a multiplication "o" and a unary operation "*" on S as
follows: for any (i, x, A) e 5 a and (j, y, /x) G 5/?,
(i, a;, A) o (j, y, /x) = ([£„'f^ * ff^f] , xy, \j]„„a Vg'a's"
1) >
(i,£, A)* = (i, lxa, A).
Then, (
-
20 Y. CHEN, Y. HE and K. P. SHUM
5 contained in E(S). Thus, by Corollary 4.4, we have (5 , * )=
[F;(Sa , *")] G VOM. It follows from Lemma 4.6(h) that the
projection of (5 , *) onto (T, + ) is indeed a unary endomorphism
with 7 as its kernel.
Conversely, let (5 , * )= [Y;(Sa, * and [Y;(Sa, *")] is the
unary semilattice decomposition of (5 , *) induced by V, we have Ia
n Ip = Aa n A^ = Ta n Tp = whenever a / j3 in F . Furthermore, we
also have
(4.4) (Va G S)(\/a G F) 07' e fa « R„ e Ia » I „ e Aa « a e 5 a
.
For any a G F , let Sa=IaxTftxAa, and set 5 = U«ey5«. Then, it
fol-lows from (4.4) that { (R a ,a^, L a ) \ a G 5} ^ 5.
Conversely, for any (r, t, I) G 5 a , by (4.4) again, there exist a
= (i, x, A), b = (j, y, /x) and c = (k, z, v) in Sa such that
(Ra,bj\Lc) = (r,t,l). Then, by virtue of Lemma 4.6 (i) and (ii), we
claim that (r,t,l)=(Ra*bc*, (a*bc*)j\ La*bc*). This implies that Sa
= { (R a ,a^,L a ) I a G So] , so that 5 = { (R a ,a^,L a ) \ a G
5} . Conse-quently, by Lemma 4.6 (iii), the mapping
a : 5 —> 5 , a 1—> (-Ra , er^, 7 a ).
is a bijection such that Saa = Sa. For any n ^ a in F and a = (
i , a;, A) G Sa, since £ is a right congruence
and TZ is a left congruence on 5, we can define i£fK G T* (7K)
and r ? ^ G T ( A K ) as follows:
£ O K ( - ^ C ) = Rac, Lcrf^K = Lca (c G SK).
For any (I, z, 0) G 5 a , we have
£,aaa(R(l,z,o)) = R(i,x,X)(l,z,o) = R(i,xz,o) = Ra,
L (l,z,o)Vaaa = £(2,z,o)(i,ii:,A) = Ly^x) = L a .
Thus £^a =[Ra] and rf^a =[La]. Assume that b = (j,y,/j.) G Sp
such that ab = (k, w, v). Take (I, z, 0) G Sap. Then we have
Caaa/3 * Cpaals( R(l,z,o) ) = R(i,x,\)(j,y,^)(l,z,o) =
R(k,wz,K)
L (l,z,o)Vaaal3Vl3aal3 = L (l,z,o)(i,x,X)(j,y,n) = ^{l,zw,v) = L
ab-
Acta Mathematica Hungarica, 2008
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PROJECTIVELY CONDENSED SEMIGROUPS 21
Consequently, we conclude that &aa/3 * ̂a
a/3 =[Rab] and r]^a/3T]b^a/3 =[Lab].
Thus, the set S forms a groupoid under the multiplication denned
by
aaoba = (Rab, cr^ • bj\Lab) = (ab)a.
Obviously, a is a groupoid isomorphism which maps from S onto S.
Thus S is a semigroup and a is a semigroup isomorphism. It follows
that, for
any K ^ a[3 in Y, £^8° = £OK * £/TK an
-
22 Y. CHEN, Y. HE and K. P. SHUM
Therefore, (S, *) e VSeSuAb follows by Lemma 3.6. (ii) =>
(iii). Assume that (S, *) e P5e5u^6 satisfying the condition
(CR).
Since C7 is a subband of S, by Lemma 3.6, we have
(5.1) (VxeS) (x,x*)enu.
Let (a, 6) e T̂ ^7 and c e. S. By the dual result of Lemma 1.1
(ii) and (5.1),
we derive that (a*,b*) e Hu\u = TZ\V. Furthermore, since the
Green's rela-
tion TZ on S is a left congruence contained in 1ZU, we also
have
(5.2) (ca*,cb*) G 1Z Q 1ZU.
The fact c* e [7̂ c* = [/(racH!)* implies (ac*)Vac* = ac*.
Consequently, we
have c*ac* C ac* since c*ac* = c* • ac*. Similarly, we also have
c*ac* TZ c*a. Since (S,*) satisfies the condition (CR) and
(a,a*),(c,c*) e Hu g Cu', we obtain the following relations in
S:
(5.3) ca Cu c*a 1Z c*ac* C ac* Cu a*c*, c*a* Cu ca*.
Consequently, by Lemma 1.1 (i) and its dual result, we get
ca Cu c*a 1ZU c*ac* Cu ac* Cu a*c*, c*a* Cu ca*,
and thus
(5.4) (ca)* Zu (c*a)* Uu (c*ac*)* Zu (ac*)* Zu a*c*, c*a* Zu
(ca*)*.
Moreover, by virtue of (5.4), Lemma 1.1 and its dual result,
(5.5) (ca)* C(U) (c*a)* K(U) (c*ac*)* C(U) (ac*)* C(U) a*c*,
c*a* C(U) (ca*)*.
Then, it follows from the fact c*a* V(U) a*c* that
(5.6) (ca)* V(U) a*c* V(U) c*a* V(U) (ca*)*.
We have proved that (ca)* and (ca*)* are in the same maximal
rectangular subband of U. On the other hand, since
(ca*)*ca = (ca*)* • c • a*a = (ca*)* • ca* • a = ca* • a =
ca,
Acta Mathematica Hungarica, 2008
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PROJECTIVELY CONDENSED SEMIGROUPS 2 3
we have (ca*)* G Ulca = U\cayj whence
(5.7) (ca*)*(ca)* = (ca)*.
Bv (5 6) and (5 7) ((ca*)* (ca)*) eH(U) whence ((ca*)* (ca)*) G
7£ c 7 ^ in S. Thereby
(5.8) ca* Uu (ca*)* Uu (ca)* Uu ca.
A similar argument shows that
(5.9) cb* TZU cb.
By summarizing (5.2), (5.8) and (5.9), we claim that (ca,cb) G
1ZU. Thus (S, *) indeed satisfies the condition (CL).
(iii) => (i). Assume that (S, *) G VSeSuAb satisfies the
condition (C).
For any a, b G S, since (a, a*), (b, b*) e Hu = Zu n TZU, we
have the relation ab Cu a*b TZU a*b*, and thus (ab)* Cu (a*b)* TZU
a*b*. Furthermore, by Lemma 1.1 (ii) and its dual result, we can
see that
(ab)* C(U) (a*b)* K(U) a*b*,
and so (ab)* V(U) a*b*. Consequently, (S, *) e VOM follows by
Lemma 4.1. •
REMARK 5.2. By Theorem 5.1 and Theorem 3.1,
VOM = VOM n VSeSuAb = VOM n StVSeKpp = VOM n StVSeCpp,
VOM nC = VOM n SeSuAb = VOM nStSeKpp = VOM nStSeCpp,
VOM n StKpp = VOM n SuKpp
= { (5 , *) G SuTZpp | E(S) is a subsemigroup of 5} ,
PCM n stCpp = VOM n Su£pp
= { (S, *) G SuCpp | E(S) is a subsemigroup of S} ,
VOM n SuAb = { (5, *) G SuAb \ E(S) is a subsemigroup of 5}
.
In what follows, we denote the quasivarieties VOM n C, POTW n
SuKpp, VOM n Su£pp and VOM n 5u^6 by 0.M, OSuKpp, OSuCpp and C5u^6,
respectively. By the above equations, we can see that VOM, OM,
OSuKpp, OSuCpp and OSuAb are all of the quasivarieties of
generalized orthogroups. Clearly, with respect to inclusion
relation, these quasivarieties form a semi-lattice with the
following Hasse diagram:
Acta Mathematica Hungarica, 2008
-
24 Y. CHEN, Y. HE and K. P. SHUM
\
OSuAb
Fig. 3
REMARK 5.3. (i) If T is a semilattice Y of unipotent monoids,
then we can easily show that E(T) forms a subsemigroup of T. Thus,
by Remark 4.5 and Lemma 1.1 (iii), the following assertion is
evident: a semigroup T is in C n CPOM if and only if T is a
semilattice of unipotent monoids.
(ii) The rpp semigroups with central idempotents are called
C-rpp semi-groups (see [2]). By Remark 4.5, SuRpp n CPOM = StRpp n
CPOM is a sub-class of the class of C-rpp semigroups. Conversely,
for any C-rpp semi-group T, since E(T) is a semilattice and Q = C*
on T, by Corollary 3.5, T G SuRpp n CPOM. Thus, SuRpp n CPOM is
exactly the class of C-rpp semi-groups.
THEOREM 5.4. Let (S,*) = [ Y;Ta;Ia,Aa;^'X),rg'A) ] G VOM.
Then
(i) (S, *) G OM if and only if each Ta is a unipotent monoid; I
V VI VAI I V\M \S I VV Kf VI l^Wlljl V (ii) (S, *) G OSuKpp if and
only if each Ta is a left cancellative monoid
and, for any [3 ^ a in Y and (i, x, A) G Sa, ry^'Z' and ry^'Z'
are C-equiv-
alent in T(Ap); (iii) (S,*) G OSuCpp if and only if each Ta is a
right cancellative monoid
and, for any (3^ainY and (i, x, A) G Sa, £^'Z' and
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PROJECTIVELY CONDENSED SEMIGROUPS 25
We omit the details, (iii) is the dual of (ii). By using
Corollary 3.9 and the above statements (ii) and (iii), we get (iv).
•
Acknowledgement. The authors thank the referee for suggestions
and corrections.
[1
[2
[3;
[4
[5 [e;
[7
[«:
[9
[io;
[11
[12:
[13 [14
[15
[is;
[17;
[is;
[19
[20
[21
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