Regular semi-groups whose idempotents satisfy permutation identities

Yamada, Masashi

doi:10.2140/pjm.1967.21.371

Cited by 75 publications

(51 citation statements)

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“…In [24] we already made the remark that every normal band divides a band which is the direct product of a semilattice and a rectangular band; moreover, every orthodox pseudo-inverse semigroup S (i.e. generahzed inverse semigroup [48]) divides a pseudo-inverse semigroup which is the direct product of an inverse semigroup and a rectangular band [24,Theorem 4.8]. Every fundamental pseudo-inverse semigroup 5 divides a fundamental pseudo-inverse semigroup which is an elementary rectangular band of fundamental inverse semigroups [24, Theorem 4.2].…”

Section: Resultsmentioning

confidence: 99%

The structure of pseudo-inverse semigroups

Pastijn¹

1982

Trans. Amer. Math. Soc.

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Abstract. A regular semigroup S is called a pseudo-inverse semigroup if eSe is an inverse semigroup for each e = e1 G S. We show that every pseudo-inverse semigroup divides a semidirect product of a completely simple semigroup and a semilattice. We thereby give a structure theorem for pseudo-inverse semigroups in terms of groups, semilattices and morphisms. The structure theorem which is presented here generalizes several structure theorems which have been given for particular classes of pseudo-inverse semigroups by several authors, and thus contributes to a unification of the theory.Completely (0-) simple semigroups and inverse semigroups form the first prototypes for the study of pseudo-inverse semigroups. We therefore can say that the theory of regular semigroups began with the study of pseudo-inverse semigroups [40,45]. "We may distinguish four successful trends in the papers which since then have dealt with some wider classes of pseudo-inverse semigroups: 1. the subdirect products of completely 0-simple and completely simple semigroups, 2. the generalized inverse semigroups (orthodox pseudo-inverse semigroups, 3. the normal band compositions of inverse semigroups, and 4. Rees matrix semigroups over inverse semigroups (with zero).Subdirect products of completely 0-simple semigroups and completely simple semigroups were initiated in [13, Chapter 2] and studied in great detail in [18] (see also §4 of [14]); this class contains several interesting subclasses: (a) the trees of completely 0-simple semigroups [18] which include the primitive regular semigroups [7, Vol. II,16,39,44,46] Key words and phrases. Pseudo-inverse semigroup, proper inverse semigroup, rectangular band of inverse semigroups, pseudo-semilattice, semilattice, order automorphism.'The author's research was done while he was a visiting professor at the University of Nebraska, supported by a Fulbright-Hays Award. [8,10,49] which of course yield special inverse semigroups. The idea of a Rees matrix representation has been exploited and applied to produce numerous classes of sophisticated semigroups; we refer the reader to [17, 35, §4], for the peculiar pseudo-inverse semigroups which have a structure theorem of Rees type. The above considered classes of pseudo-inverse semigroups may overlap. However, so far no attempt has been made to establish a comprehensive classification.Independently from the above-mentioned cases several devices have been invented to build pseudo-inverse semigroups [1, Chapter 4, 2,19,25,47...]. Some recent papers concentrate on idempotent-generated nonprimitive pseudo-inverse semigroups [2,3, 4,15,24,34], and in [32] a countably infinite set of pairwise nonisomorphic bisimple nonprimitive pseudo-inverse semigroups with 3 idempotent generators has been constructed.The class of pseudo-inverse semigroups was introduced in [29 and 30] as an overall generalization of the specific classes listed above. The structure theorem for pseudo-inverse semigroups, which is given in [29], presupposes the knowledge of the biordered set, ...

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Section: Resultsmentioning

confidence: 99%

The structure of pseudo-inverse semigroups

Pastijn¹

1982

Trans. Amer. Math. Soc.

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“…A semigroup S is a generalized inverse semigroup (Yamada [16]) if it is orthodox and the idempotents of S form a normal band. It follows immediately from Lemma 3.2 and identity (V) that any strict *-semigroup is a generalized inverse semigroup.…”

Section: If E Is An Idempotent Of a Strict *-Semigroup Then E* = Ementioning

confidence: 99%

“…In this section we develop a structure theorem for strict *-semigroups in terms of rectangular bands and inverse semigroups. The construction used is quite similar to that introduced by Yamada [16].…”

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confidence: 99%

The join of the varieties of strict inverse Semigroups and rectangular bands

Petrich

Reilly

1984

Glasgow Math. J.

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1. Introduction and summary. In recent years, certain varieties of semigroups with unary operations (of "inversion") have received considerable attention. Generally speaking, these have been contained in one or other of the two classes of completely regular semigroups (that is, semigroups that are unions of groups) and inverse semigroups. The varieties of all completely regular semigroups and of all inverse semigroups have relatively little in common, in that their intersection is simply the variety of all semilattices of groups. As a result, the investigations have been conducted independently with no interplay between the two theories.The purpose of this note is to move towards viewing completely regular semigroups and inverse semigroups as members of the variety of unary semigroups and to study classes of unary semigroups with larger intersections with completely regular semigroups and inverse semigroups than merely semilattices of groups. As a first step in this programme, we consider here the join of the variety of rectangular bands and the variety of inverse semigroups generated by Brandt semigroups (that is, the variety of strict inverse semigroups) as varieties of unary semigroups. Equivalently (as we shall show) we consider the join of the variety of orthodox normal bands of groups and the variety of strict combinatorial inverse semigroups.In this way we break out from the lattices of varieties of completely regular semigroups and of inverse semigroups and initiate the investigation of the joins of their subvarieties. This study opens up new vistas in the theory of varieties of unary semigroups.The second section brings together important background information. In Section 3, we define the variety of strict *-semigroups if*', develop some of its elementary properties and give relevant examples. Section 4 introduces two important equivalence relations T and T' on strict ^-semigroups which are subsequently used to derive structurally important rectangular bands from any strict ^-semigroup. The intersection of T, T' and the minimum inverse semigroup congruence y is also studied here.In Section 5, it is shown that any strict *-semigroup is a subdirect product of rectangular bands, groups and 0-direct products of rectangular bands with Brandt semigroups. From this representation theorem is derived the main result: 5^* = 9iS9vS8v^, where 9?39, 98 and 'S denote, respectively, the varieties of rectangular bands, combinatorial strict inverse semigroups and groups.

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“…The equivalence of (iii) and (iv) was proved by Yamada [12, Theorem 4] and, in a slightly different form, by the author [9]. An extensive use of spined products for various subclasses of these semigroups can be found in [13]. The implication " (iii) implies (iv) " can also be obtained by specializing the main theorem in [4]; however, the above proof makes the use of the hypotheses much more transparent and represents a basis for several proofs in the succeeding sections.…”

Section: Sx={(eg)ee S Xsi£\n T T = N T }mentioning

confidence: 99%

Regular semigroups which are subdirect products of a band and a semilattice of groups

Petrich

1973

Glasgow Math. J.

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1. Introduction and summary. In the study of the structure of regular semigroups, it is customary to impose several conditions restricting the behaviour of ideals, idempotents or elements. In a few instances, one may represent them as subdirect products of some much more restricted types of regular semigroups, e.g., completely (0-) simple semigroups, bands, semilattices, etc. In particular, studying the structure of completely regular semigroups, one quickly distinguishes certain special cases of interest when these semigroups are represented as semilattices of completely simple semigroups. In fact, this semilattice of semigroups may be built in a particular way, idempotents may form a subsemigroup, Jf may be a congruence, and soon.Instead of making some arbitrary choice of these conditions, we consider, in a certain sense, the converse problem, by starting with regular semigroups which are subdirect products of a band and a semilattice of groups. This choice, in turn, may seem arbitrary, but it is a natural one in view of the following abstract characterization of such semigroups: they are bands of groups and their idempotents form a subsemigroup. It is then quite natural to consider various special cases such as regular semigroups which are subdirect products of: a band and a group, a rectangular band and a semilattice of groups, etc. For each of these special cases, we establish (a) an abstract characterization in terms of completely regular semigroups satisfying some additional restrictions, (b) a construction from the component semigroups in the subdirect product, (c) an isomorphism theorem in terms of the representation in (b), (d) a relationship among the congruences on an arbitrary regular semigroup which yield a quotient semigroup of the type under study.Using the following abbreviations: B-bands; SG-semilattices of groups; L-left zero semigroups; R-right zero semigroups; S-semilattices; G-groups; 1-one element semigroups, we consider regular semigroup subdirect products of these according to the diagram below.This leads us to classes of semigroups for which we introduce the following abbreviations: V-a variety of bands; UVG-semigroups 5 for which $e is a congruence, S/Jfe V, and the idempotents form a unitary subset of S; M-rectangular bands; CRISN-completely regular semigroups whose idempotents form a strongly normal subband (defined in §2); CRILSNare CRISN with left added in front of" strongly "; CRUSN-are CRISN whose idempotents form a unitary subset; CRTJLSN-conjunction of the last two. In addition we adopt from [5]: ISBG-bands of groups whose idempotents form a subsemigroup. Thus, we call a congruence p on a semigroup S an A'-congruence if Sjp is in class X, e.g., for X = G, a group congruence, for X = S, a semilattice congruence (should not be confused with the letter S which usually denotes a semigroup).

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Regular semi-groups whose idempotents satisfy permutation identities

Cited by 75 publications

References 8 publications

The structure of pseudo-inverse semigroups

The structure of pseudo-inverse semigroups

The join of the varieties of strict inverse Semigroups and rectangular bands

Regular semigroups which are subdirect products of a band and a semilattice of groups

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