Profinite categories and semidirect products
type: Item
instance of: scholarly article
title: Profinite categories and semidirect products
zbMATH Open document ID: 0891.20037
DOI: 10.1016/S0022-4049(96)00083-7
published in: Journal of Pure and Applied Algebra
publication date: 1998-07-19T00:00:00Z
review text: This very substantial paper is motivated by a complementary question pair -- for a given semidirect product \({\mathbf V}*{\mathbf W}\) of pseudovarieties of semigroups when is a given pseudoidentity valid in \({\mathbf V}*{\mathbf W}\) and what is a basis of pseudoidentities for \({\mathbf V}*{\mathbf W}\)? Although the authors go through considerable length to introduce various facets of their theory: implicit operations, pseudovarieties of categories, the notion of a local pseudovariety of monoids, and relational morphisms (which form an ingredient of a certain extension of the idea of semidirect product), the paper is largely an attempt to extend Tilson's Theory of Derived Categories by providing what the authors see as the `missing ingredient': a Reiterman type theorem for pseudovarieties of categories (Theorem 2.7) which admits a short proof based on the projective limit of profinite categories. The authors use their extended semidirect product to produce a description of a basis of pseudo-identities for a given semidirect product which, although not effective, provides membership criteria in many interesting cases. A generalisation of Tilson's Delay Theorem is given along with some consequences and applications: for instance the lattice of pseudovarieties of categories is naturally embedded in the lattice of pseudovarieties of semigroups.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 1121660
zbMATH Keywords: derived categories
zbMATH Keywords: bases of pseudoidentities
zbMATH Keywords: semidirect products
zbMATH Keywords: pseudovarieties of semigroups
zbMATH Keywords: implicit operations
zbMATH Keywords: pseudovarieties of categories
zbMATH Keywords: local pseudovarieties of monoids
zbMATH Keywords: relational morphisms
zbMATH Keywords: profinite categories
zbMATH Keywords: lattices of pseudovarieties
author: Jorge Almeida
author: Pascal Weil
reviewed by: Peter M. Higgins
MaRDI profile type: Publication
cites work: Semidirect products of pseudovarieties from the universal algebraist's point of view
cites work: Implicit operations on finite \({\mathcal J}\)-trivial semigroups and a conjecture of I. Simon
cites work: Q4311850
cites work: A SYNTACTICAL PROOF OF LOCALITY OF DA
cites work: Q4296426
cites work: Free profinite semigroups over semidirect products
cites work: Q3123634
cites work: Q3772420
cites work: Q3807430
cites work: The Birkhoff theorem for varieties of finite algebras
cites work: On generalized locally testable languages
cites work: Characterizations of locally testable events
cites work: Q4132170
cites work: Q3794424
cites work: Profinite categories, implicit operations and pseudovarieties of categories
cites work: Q3673124
cites work: Q4271190
cites work: Inverse semigroups and extensions of groups by semilattices
cites work: Inverse semigroups and varieties of finite semigroups
cites work: On a conjecture of Rhodes
cites work: The Birkhoff theorem for finite algebras
cites work: The kernel of monoid morphisms
cites work: Decomposition techniques for finite semigroups, using categories. II
cites work: THE PRO-p TOPOLOGY OF A FREE GROUP AND ALGORITHMIC PROBLEMS IN SEMIGROUPS
cites work: Finite semigroup varieties of the form V*D
cites work: Graph congruences and wreath products
cites work: Categories as algebra: An essential ingredient in the theory of monoids
cites work: Closure of varieties of languages under products with counter
full work available at URL: s0022-4049(96)00083-7
OpenAlex ID: W2104134222
Wikidata QID: Q126778478
description: scientific article; zbMATH DE number 1121660