On finite generation and presentability of Schützenberger products

Abstract

The finite generation and presentation of Schutzenberger products of semigroups are investigated. A general necessary and sufficient condition is established for finite generation. The Schutzenberger product of two groups is finitely presented as an inverse semigroup if and only if the groups are finitely presented, but is not finitely presented as a semigroup unless both groups are finite.2000 Mathematics subject classification: primary 20M05; secondary 20M18.

Generators and presentations for direct and wreath products of monoid acts

“…Continuing in this fashion (and recalling that A is finite), we deduce that θ is (U, a)-connected to c aθ . Therefore, in any of the cases (1), (2), (3) and (4), we have that A ≀ B is finitely presented by Proposition 5.5.…”

“…Diagonal acts were first mentioned, implicitly, in a problem in the American Mathematical Monthly [1], and have since been intensively studied by several authors (see [3], [4], [14]). A systematic study of finite generation of diagonal acts was undertaken by Gallagher in his PhD thesis [2]. He showed that infinite monoids from various 'standard' monoid classes, such as commutative, inverse, idempotent, cancellative, completely regular and completely simple, do not have finitely generated diagonal acts (see [3]).…”

Semigroups for which every right congruence of finite index is finitely generated

“…We note that S X is a subsemigroup of both Surj X and Inj X . It was shown in the proofs of [6,Theorems 4.4.2 and 4.4.4] that Surj X \S X is an ideal of Surj X and Inj X \S X is an ideal of Inj X . Therefore, since S X is not f-noetherian, we have that Surj X and Inj X are not f-noetherian by Lemma 2.17.…”

On Generators and Presentations of Semidirect Products in Inverse Semigroups

“…Lavers gives a presentation for general products of monoids in [4] and investigates when general products of finitely presented monoids are finitely presented. In [3] finite generation and presentability of Schützenberger products are investigated. These constructions share the property that an action of one of the building blocks is defined on the other component.…”