Semigroup expansions using the derived category, kernel, and Malcev products
type: Item
instance of: scholarly article
title: Semigroup expansions using the derived category, kernel, and Malcev products
zbMATH Open document ID: 0936.20046
DOI: 10.1016/S0022-4049(97)00172-2
author: Gillian Elston
published in: Journal of Pure and Applied Algebra
publication date: 2000-05-17T00:00:00Z
review text: For each finite set \(A\), let \({\mathcal M}_A\) denote the category of \(A\)-generated monoids, together with \(A\)-preserving monoid morphisms. An ``expansion'' is a functor from \({\mathcal M}_A\) into itself, together with a natural transformation from the functor to the identity functor such that its arrows are surjective morphisms. The purpose of this paper is to provide several constructions of expansions, based on wreath and Malcev products, both for monoids and for semigroups. For simplicity and brevity, we focus on the wreath, monoidal expansion in this review. Let \(\mathcal V\) be a semigroup variety. Given an \(A\)-generated monoid \(S\) and the associated morphism \(\phi\) from the free monoid \(A^*\) on \(A\) to \(S\), we may associate the derived category \(D_\phi\) (for which the reader is referred to \textit{B. Tilson} [J. Pure Appl. Algebra 48, 83-198 (1987; Zbl 0627.20031)]), which in this situation is isomorphic to the free category on the Cayley graph of \(S\). Let \(D\) be the greatest quotient of \(D_\phi\) whose local monoids belong to \(\mathcal V\) and let \(D_{cd}\) be its consolidation -- the semigroup obtained in the natural way from the arrows of the category by setting undefined products, if any, equal to an adjoined zero. Now, apparently motivated by the ``Derived Category Theorem'' of Tilson [op. cit.], there is an \(A\)-generated subsemigronp \(S_{\mathcal V}\) of the wreath product of \(D_{cd}\) with \(S\) which, together with the projection \(\epsilon_S\) upon \(S\), serves as the basis for the expansion. It has the property that it is the ``largest'' monoid with the property that there is a morphism upon \(S\) in whose derived category the local monoids all belong to \(\mathcal V\). In addition, it preserves various finiteness properties. A related semigroup expansion is defined. It turns out that in the case where \(\mathcal V\) is the variety of one-element semigroups, this construction yields the well-known Karnofsky-Rhodes …
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 18B40
zbMATH DE Number: 1308323
zbMATH Keywords: monoid morphisms
zbMATH Keywords: functors
zbMATH Keywords: natural transformations
zbMATH Keywords: expansions
zbMATH Keywords: Malcev products
zbMATH Keywords: semigroup varieties
zbMATH Keywords: free monoids
zbMATH Keywords: derived categories
zbMATH Keywords: wreath products
zbMATH Keywords: local monoids
zbMATH Keywords: Rhodes expansions
reviewed by: Peter R. Jones
MaRDI profile type: Publication
cites work: Almost finite expansions of arbitrary semigroups
cites work: Q4132170
cites work: Prime decomposition theorem for arbitrary semigroups: General holonomy decomposition and synthesis theorem
cites work: Q4126563
cites work: Monoid varieties defined by \(x^{n+1}=x\) are local
cites work: SEMIGROUPS WITH IDEMPOTENT STABILIZERS AND APPLICATIONS TO AUTOMATA THEORY
cites work: E-unitary inverse monoids and the Cayley graph of a group presentation
cites work: Infinite iteration of matrix semigroups. II: Structure theorem for arbitrary semigroups up to aperiodic morphism
cites work: MONOIDS ACTING ON TREES: ELLIPTIC AND WREATH PRODUCTS AND THE HOLONOMY THEOREM FOR ARBITRARY MONOIDS WITH APPLICATIONS TO INFINITE GROUPS
cites work: The kernel of monoid morphisms
cites work: Decomposition techniques for finite semigroups, using categories. II
cites work: On the equation \(x^ t=x^{t+q}\) in categories
cites work: Graph congruences and wreath products
cites work: On the complexity of finite semigroups
cites work: Categories as algebra: An essential ingredient in the theory of monoids
description: scientific article; zbMATH DE number 1308323