The variety of commutative additively and multiplicatively idempotent semirings
type: Item
instance of: scholarly article
title: The variety of commutative additively and multiplicatively idempotent semirings
zbMATH Open document ID: 1393.16039
author: Ivan Chajda
author: Helmut Länger
published in: Semigroup Forum
publication date: 2018-06-22T00:00:00Z
review text: The semirings in the title are algebras \((S,+,\cdot)\) with two semilattice structures \((S,+)\) and \((S,\cdot)\), where multiplication distributes over addition. They are quite well known under the name of meet-distributive bisemilattices, or bisemilattices with one distributive law. Several results from a long list of references are related to the results of the paper. Let \(V\) be the variety of meet-distributive bisemilattices. The (5-element) lattice of all subvarieties of \(V\) was fully described in [\textit{R. McKenzie} and the reviewer, Proc. Klagenfurt Conf. 1978, 213--218 (1979; Zbl 0419.06003)]. Among other things, this paper contains ``(a) a normal form for terms'' and a remark (a direct consequence of the main result) that ``(b) the variety \(V\) is generated by any one subdirectly irreducible algebra which is neither a lattice nor a semilattice.'' Free algebras in \(V\) were fully described in [the reviewer, Demonstratio Math. 13, 565--572 (1980, Zbl 0048.08006)]. All subdirectly irreducible \(V\)-algebras with at least one semilattice reduct a chain were characterised in [the reviewer, Algebra Universalis 10, 36--47, (1980, Zbl 0434.06005)]. A representation theorem for meet-distributive bisemilattices was given in [the reviewer and \textit{J. D. H. Smith}, J. Algebra 70, 78--88 (1981; Zbl 0457.06002)]. In the paper under review, the authors consider the same algebras with additional constants \(1\) as a neutral element for the multiplication, and \(0\) as both a zero of the multiplication and a neutral element for the addition. They describe some subdirectly irreducible algebras with both reducts being chains, and derive (a) and (b) (for those subdirectly irreducible algebras).
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 06A12
Mathematics Subject Classification ID: 08B20
Mathematics Subject Classification ID: 08B26
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 6893043
zbMATH Keywords: semiring
zbMATH Keywords: commutative
zbMATH Keywords: additively idempotent
zbMATH Keywords: multiplicatively idempotent
zbMATH Keywords: variety
zbMATH Keywords: locally finite
zbMATH Keywords: residually large
zbMATH Keywords: word problem
Wikidata QID: Q59607519
reviewed by: Anna Romanowska
MaRDI profile type: Publication
full work available at URL: s00233-017-9905-2
OpenAlex ID: W2770533145
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DOI: 10.1007/S00233-017-9905-2
description: scientific article; zbMATH DE number 6893043