Perfect semigroups (original) (raw)
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Proceedings of the Edinburgh Mathematical Society, 1979
A monoid in which every principal right ideal is projective is called a right PP monoid. Special classes of such monoids have been investigated in (2), (3), (4) and (8). There is a well-known internal characterisation of right PP monoids using the relation ℒ* which is defined as follows. On a semigroup S, (a,b) ∈ℒ* if and only if the elements a,b of S are related by Green's relation ℒ* in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each ℒ*-class of S contains an idempotent. The existence of an identity element is not relevant for the internal characterisation and in this paper we study some classes of semigroups whose idempotents commute and in which each ℒ*-class contains an idempotent. We call such a semigroup a right adequate semigroup since it contains a sufficient supply of suitable idempotents. Dually we may define the relation ℛ* on a semigroup and the notion of a left adequate semigroup. A semigroup which is both left and right adequate w...
Inverse semigroups with zero: covers and their structure
Journal of the Australian Mathematical Society, 1999
We obtain analogues, in the setting of semigroups with zero, of McAlister's covering theorem and the structure theorems of McAlister, O'Carroll, and Margolis and Pin. The covers come from a class C of semigroups defined by modifying one of the many characterisations of E-unitary inverse semigroups, namely, that an inverse semigroup is E-unitary if and only if it is an inverse image of an idempotent-pure homomorphism onto a group. The class C is properly contained in the class of all £*-unitary inverse semigroups introduced by Szendrei but properly contains the class of strongly categorical £*-unitary semigroups recently considered by Gomes and Howie. 1991 Mathematics subject classification (Amer. Math. Soc): primary 20M18.
On Finite Complete Presentations and Exact Decompositions of Semigroups
Communications in Algebra, 2011
We prove that given a finite (zero) exact right decomposition (M, T) of a semigroup S, if M is defined by a finite complete presentation then S is also defined by a finite complete presentation. Exact right decompositions are natural generalizations to semigroups of coset decompositions in groups. As a consequence we deduce that the Zappa-Szép extension of a monoid defined by a finite complete presentation, by a finite monoid is also defined by such a presentation. It is also shown that when a semigroup A isomorphic to a variant semigroup A(x) that is defined by a finite complete presentation, where x belongs to a sandwich matrix P , together with some other conditions, we deduce that the zero Rees matrix semigroup M 0 [A; I, J; P ] is also defined by a finite complete presentation.
Semigroups with finitely generated universal left congruence
Monatshefte für Mathematik
We consider semigroups such that the universal left congruence ω is finitely generated. Certainly a left noetherian semigroup, that is, one in which all left congruences are finitely generated, satisfies our condition. In the case of a monoid the condition that ω is finitely generated is equivalent to a number of pre-existing notions. In particular, a monoid satisfies the homological finiteness condition of being of type left-FP 1 exactly when ω is finitely generated. Our investigations enable us to classify those semigroups such that ω is finitely generated that lie in certain important classes, such as strong semilattices of semigroups, inverse semigroups, Rees matrix semigroups (over semigroups) and completely regular semigroups. We consider closure properties for the class of semigroups such that ω is finitely generated, including under morphic image, direct product, semi-direct product, free product and 0-direct union. Our work was inspired by the stronger condition, stated for monoids in the work of White, of being pseudo-finite. Where appropriate, we specialise our investigations to pseudo-finite semigroups and monoids. In particular, we answer a question of Dales and White concerning the nature of pseudo-finite monoids. Keywords Monoids • Semigroups • Left congruences • Finitely generated • FP 1 • Pseudo-finite Communicated by J. S. Wilson.