Embedding theorems for proper inverse semigroups

O’Carroll, Liam

doi:10.1016/0021-8693(76)90023-5

Cited by 39 publications

(27 citation statements)

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“…One of the fundamental results in the structure theory of semigroups is that every inverse semigroup is an idempotent separating homomorphic image of a 'proper' inverse semigroup where a proper inverse semigroup is isomorphic to a so called 'P-semigroup' and embeds into a semidirect product of a semilattice by a group [17][18][19]. For (left) restriction semigroups, many authors have produced work using constructions similar (at least on the surface) to those of McAlister, replacing groups by monoids of various kinds.…”

Section: It Is Easy To See Thatmentioning

confidence: 99%

Algebraic properties of Zappa–Szép products of semigroups and monoids

Zenab

2017

Semigroup Forum

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Direct, semidirect and Zappa-Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa-Szép products of monoids and semigroups and investigate generalised Greens relations R * , L * , R E and L E for these Zappa-Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa-Szép product E S. We further investigate properties of E S and show that S is a retract of E S. We also find a subset T of E S which is left restriction.

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Section: It Is Easy To See Thatmentioning

confidence: 99%

Algebraic properties of Zappa–Szép products of semigroups and monoids

Zenab

2017

Semigroup Forum

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“…We now present a theorem which is an application of our Lemma 2, and which is a refinement for the results mentioned here. It should be remarked that the following theorem could also be obtained using Lemma 1.2 of O'Carroll (1976), together with the results of McAlister (1974a, b). Our construction, however, has the advantage that the proofs are purely semilattice-theoretic and independent of any knowledge about the construction of .Zf-unitary inverse semigroups.…”

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confidence: 93%

“…The particular procedure which will be given here is very likely to be relevant for the construction of inverse semigroups. The reader may consult McAlister (1974aMcAlister ( , b, 1978 and O'Carroll (1976) for a further motivation of our embedding theorem.…”

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confidence: 99%

Semilattices with a transitive automorphism group

Pastijn

1980

J Aust Math Soc A

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If/, is any semilattice, let T L denote the Munn semigroup of L, and Aut (L) the automorphism group of L.We show that every semilattice L can be isomorphically embedded as a convex subsemilattice in a semilattice V which has a transitive automorphism group in such a way that (i) every partial isomorphism a of L can be extended to an automorphism a. of L', (ii) every partial isomorphism a: eL -*fL of L can be extended to a partial isomorphism We shall give a procedure for embedding any semilattice in a semilattice which has a transitive automorphism group. The particular procedure which will be given here is very likely to be relevant for the construction of inverse semigroups. The reader may consult McAlister (1974aMcAlister ( , b, 1978 and O'Carroll (1976) for a further motivation of our embedding theorem. We follow the terminology and the notation of Birkhoff (1967) andHowie (1976). If L,A is any semilattice, and eeL, then the principal ideal of L which is generated by e will be denoted by eL. If e and / are elements of L such that a: eL^-fL is an isomorphism, then a will be called a partial isomorphism of L. The partial isomorphisms of L form an inverse subsemigroup T L of the symmetric

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“…We refer the reader to [21,22,31] for more information concerning P-semigroups. In case L -% is trivial, Result 5 reduces to the well-known structure theorem for completely simple semigroups.…”

Section: The Mappingsmentioning

confidence: 99%

“…Such a semigroup will be called an elementary rectangular band of inverse semigroups if for all (/, X), (j, p) E I X A we have SjXSj^ = S^. A proper inverse semigroup is an inverse semigroup which has injective structure mappings [21,22,31]. §4 of [35] gives a structure theorem for elementary rectangular bands of proper inverse semigroups.…”

Section: The Mappingsmentioning

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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Embedding theorems for proper inverse semigroups

Cited by 39 publications

References 9 publications

Algebraic properties of Zappa–Szép products of semigroups and monoids

Algebraic properties of Zappa–Szép products of semigroups and monoids

Semilattices with a transitive automorphism group

The structure of pseudo-inverse semigroups

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