Irreducible elements and uniquely generated algebras
type: Item
instance of: scholarly article
title: Irreducible elements and uniquely generated algebras
zbMATH Open document ID: 1001.08001
DOI: 10.1016/S0012-365X(01)00364-8
published in: Discrete Mathematics
publication date: 2002-05-15T00:00:00Z
review text: This paper deals with unique generation of algebras in a variety \(V\). An algebra \(A\) is uniquely generated by a set \(G\) if \(G\) generates \(A\) and \(G\) is a subset of every set that generates \(A\). The following situations are investigated: (1) every algebra in \(V\) is uniquely generated; (2) every finite algebra in \(V\) is uniquely generated; (3) every free algebra in \(V\) is uniquely generated; (4) every nontrivial algebra in \(V\) is uniquely generated. In section 2, the authors provide instances of special properties that a variety has, if the free algebras in the variety are uniquely generated. Theorem 3.1 provides a classification of the subvarieties of the variety of commutative semigroups according to the four situations mentioned above. Section 4 provides partial answers to these questions for Ockham and De Morgan algebras. In particular, all finite De Morgan algebras that are uniquely generated are determined. Section 7 provides a listing of distinct varieties generated by two-element algebras, grouped according to which of the algebras in the variety are uniquely generated. Irreducible elements are used as a primary tool in the investigations. The paper is fairly self-contained modulo elementary knowledge of universal algebra.
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 08A05
Mathematics Subject Classification ID: 08B20
zbMATH DE Number: 1741678
zbMATH Keywords: uniquely generated algebra
zbMATH Keywords: irreducible element
zbMATH Keywords: inaccessible element
zbMATH Keywords: commutative semigroups
author: Joel Berman
author: Gabriela Hauser-Bordalo
reviewed by: V. S. Ramamurthi
MaRDI profile type: Publication
full work available at URL: s0012-365x(01)00364-8
OpenAlex ID: W1997985397
description: scientific article; zbMATH DE number 1741678