projectively condensed semigroups, generalized completely...

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]

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

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


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



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


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.



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.



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



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.


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 (

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


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})°



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*. •



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


£,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



(S,*). By the notation (S,*)= [Y;(Sa,*")] G VOM, we always consider that (S, *) G VOM and [Y;(Sa, *


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, *


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.



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, (


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-



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


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,


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:



\

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


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.

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