projectively condensed semigroups, generalized

Acta Math. Hungar., 119 (3) (2008), 281305.

DOI: 10.1007/s10474-007-7038-x First published online January 22, 2008

PROJECTIVELY CONDENSED SEMIGROUPS, GENERALIZED COMPLETELY REGULAR SEMIGROUPS AND PROJECTIVE ORTHOMONOIDS Y. CHEN1 ∗ , Y. HE2 1

†

and K. P. SHUM3

‡

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 Fabruary 22, 2007; revised July 25, 2007; accepted August 6, 2007)

Abstract. The class PC of projectively condensed semigroups is a quasivariety of unary semigroups, the class of projective orthomonoids is a subquasivariety of PC . Some well-known classes of generalized completely regular semigroups will be regarded as subquasivarieties of PC . We give the structure semilattice composition and the standard representation of projective orthomonoids, and then obtain the structure theorems of various generalized orthogroups.

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). † Corresponding author. Partially supported by a grant of Natural Scientic Foundation of Hunan (No. 06JJ2025) and a grant of Scientic 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 Classication: 20M07, 08C15.

c 2008 Akadémiai Kiadó, Budapest 02365294/$ 20.00 °

282

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 ∈ S , the set of all [left, right] idempotent identities of a is denoted by Ia [Ial , Iar ], the intersection of U and Ia [Ial , Iar ] is denoted by Ua [Ual , Uar ]. The natural partial order 5 on E(S) is a partial order relation dened as © ª 5= (e, f ) ∈ E(S) × E(S) | f ∈ Ie . If ρ is an equivalence on S such that |ρa ∩ U | = 1 [especially, |ρa ∩ Ua | = 1] for all a ∈ S , then ¡ S is said ¢ to be [strongly] (ρ, U )-surjective. In particular, if S is [strongly] ρ, E(S) -surjective, then S is said to be [strongly] ρ-surjective. The right congruence L∗ and the equivalences Le, LeU on S are dened by © ª L∗ = (a, b) ∈ S × S | (∀x, y ∈ S 1 ) ax = ay ⇔ bx = by , © ª © ª Le = (a, b) ∈ S × S | Iar = Ibr , LeU = (a, b) ∈ S × S | Uar = Ubr .

e, R e U on S are dened dually. The left congruence R∗ and the equivalences R Let e = Le ∩ R, e e U = LeU ∩ R eU . H∗ = L∗ ∩ R∗ , H H Note that

Le = LeE(S) ,

e=R e E(S) , R

e=H e E(S) . H

The L∗ -class of S containing an element a is denoted by L∗a . The quantities ea , L eU L a and so on are dened similarly. The following basic lemma and its dual result will be used frequently. Lemma 1.1 [1, 3, 14]. Let S be a semigroup, and U a set of idempotents of S . Then the following statements hold: (i) L j L∗ j Le j LeU ; e E(S) and L| = LeU |U ; (ii) L|E(S) = L∗ |E(S) = L| U (iii) if S is an rpp semigroup (i.e., an L∗ -surjective semigroup), then Le = ∗ L ; (iv) if S is a regular semigroup, then L = L∗ = Le; (v) for any a ∈ S and e ∈ E(S), (a, e) ∈ L∗ if and only if e ∈ Iar and for all x, y ∈ S , ax = ay implies ex = ey ; e U -class in S contains at most one element of U . (vi) each H Each row in the following Table 1 gives some basic information of a class of semigroups. For example, we can see from the rst row that: A semigroup S is called an abundant semigroup if S is L∗ and R∗ -surjective. The class of abundant semigroups is denoted by Ab. For the details concerning abundant

Acta Mathematica Hungarica 119, 2008