Abstract
We consider semigroups such that the universal left congruence omega ^{ell } 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 omega ^{ell } 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-{text {FP}}_1 exactly when omega ^{ell } is finitely generated.Our investigations enable us to classify those semigroups such that omega ^{ell } 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 omega ^{ell } 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.
Highlights
A finitary condition for a class of algebras is a condition, defined in the appropriate language, that is satisfied by at least all finite members of the class
The classes of algebras we examine here are those of semigroups and monoids
We remark that left ideals of semigroups are associated with left congruences, but, unlike the case for rings, not every left congruence comes from a left ideal
Summary
A finitary condition for a class of (universal) algebras is a condition, defined in the appropriate language, that is satisfied by at least all finite members of the class. The finitary conditions for S that are the subject of this article are those of ωS being finitely generated (as a left congruence) and the stronger condition of S being pseudo-finite. We remark that many other finitary conditions have been important in the study of semigroups and monoids, naturally including the properties of being finitely generated or finitely presented [23], and other finitary conditions on the lattices of one sided congruences, for example [7,8], both of which arise from model-theoretic considerations. 4 we consider the closure properties of the class of semigroups S with ωS being finitely generated, or of S being pseudo-finite, under standard algebraic constructions: morphisms, direct products, semidirect products, free products and 0-direct unions. The set of idempotents of a semigroup S is denoted by E(S)
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