Adequate Semigroups

Glasgow Mathematical Journal, 1993

The relation ℛ* is defined on a semigroup S by the rule that ℛ*b if and only if the elements a, b of S are related by the Green's relation ℛ in some oversemigroup of S. A semigroup S is an E-semigroup if its set E(S)of idempotents is a subsemilattice of S. A left adequate semigroup is an E-semigroup in which every ℛ*-class contains an idempotent. It is easy to see that, in fact, each ℛ*-class of a left adequate semigroup contains a unique idempotent [2]. We denote the idempotent in the ℛ*-class of a by a+.

Communications in Algebra, 1997

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Glasgow Mathematical Journal, 1986

Several definitions of a semigroup of quotients have been proposed and studied by a number of authors. For a survey, the reader may consult Weinert's paper [8]. The motivation for many of these concepts comes from ring theory and the various notions of rings of quotients. We are concerned in this paper with an analogue of the classical ring of quotients, introduced by Fountain and Petrich in [3].

Journal of the Australian Mathematical Society, 1987

The class of abundant semigroups originally arose from ‘homological’ considerations in the theory of S-systems: they are the semigroup theoretic counterparts of PP-rings. Cancellative monoids, full subsemigroups of regular semigroups as well as the multiplicative semigroups of PP-rings are abundant. In this paper we investigate the properties of Rees matrix semigroups over abundant semigroups. Some of our results generalise McAlister's work on regular Rees matrix semigroups.

Semigroup Forum

John Fountain has had a remarkably successful career in semigroup theory, as a researcher, as a Ph.D. supervisor and as a mentor. Here two of his students reflect on John's mathematical life and his influence on their careers and those of others. John B. Fountain Communicated by László Márki.

Periodica Mathematica Hungarica, 2012

We examine, in a general setting, a notion of inverse semigroup of left quotients, which we call left I-quotients. This concept has appeared, and has been used, as far back as Clifford's seminal work describing bisimple inverse monoids in terms of their right unit subsemigroups. As a consequence of our approach, we find a straightforward way of extending Clifford's work to bisimple inverse semigroups (a step that has previously proved to be awkward). We also put some earlier work of Gantos into a wider and clearer context, and pave the way for further progress. Mathematics subject classification number : 20M18.

Communications in Algebra, 2009

The Ehresmann-Schein-Nambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse semigroups. Armstrong and Lawson have successively extended the ESN Theorem to the classes of ample, weakly ample and weakly E-ample semigroups. A semigroup in any of these classes must contain a semilattice of idempotents, but need not be regular. It is significant here that these classes are each defined by a set of conditions and their left-right duals. Recently, a class of semigroups has come to the fore that is a one-sided version of the class of weakly E-ample semigroups. These semigroups appear in the literature under a number of names: in category theory they are known as restriction semigroups, the terminology we use here. We show that the category of restriction semigroups, together with appropriate morphisms, is isomorphic to a category of partial semigroups we dub inductive constellations, together with the appropriate notion of ordered map, which we call inductive radiant. We note that such objects have appeared outside of semigroup theory in the work of Exel. In a subsequent article we develop a theory of partial action and expansion for inductive constellations, along the lines of that of Gilbert for inductive groupoids.

Journal of the Australian Mathematical Society, 2009

We introduce partial actions of weakly left E-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left E-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation α↦α+, where α+ is the identity map on the domain of α. We investigate the construction of ‘actions’ from such partial actions, making a connection with the FA-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schröder, and by the second author, are to be extended appropriately to the case of weakly left E-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left E-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1981

SynopsisThe investigation of general quasi-adequate semigroups is initiated. These are semigroups which are abundant and in which the idempotents form a subsemigroup. For such a semigroupSwe study the minimum good congruence γ such thatS/γis adequate. Results on γ together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi-adequate semigroups.

Communications in Algebra, 2012

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands.

Chinese Science Bulletin, 1997

SEMIGROUP S is said to be rpp (dually, lpp) if every principal right ideal US' ( S'U ) of S, regarded as a right (left) s'-system, is The semigroup which is both rpp and