Equationally complete classes of commutative monoids with monus
type: Item
instance of: scholarly article
title: Equationally complete classes of commutative monoids with monus
zbMATH Open document ID: 0549.20039
DOI: 10.1007/BF01182254
author: K. Amer
published in: Algebra Universalis
publication date: 1984-01-01T00:00:00Z
review text: Let \((M,+,0)\) be a commutative monoid such that the relation \(a\leq b\) iff \(b=a+c\) for some \(c\in M\) is a partial order on M. Suppose that for any a,\(b\in M\) there exists a smallest \(x\in M\), \(x=a\dot-b\) with \(a\leq b+x\); then M is called a commutative monoid with monus (CMM). \textit{B. Bosbach} [Math. Ann. 161, 279-295 (1965; Zbl 0154.017)] has proved that the class of all CMM's forms a variety. The results of this paper are concerned with this variety and are the following: 1) The variety \(V({\mathbb{N}})\) of CMM's generated by the natural numbers \(({\mathbb{N}},+,0)\) with \(a\dot-b=a-b\) if \(a\geq b\) and \(=0\) otherwise consists of the positive cones of arbitrary lattice ordered groups. 2) The variety \(V(2)\) of CMM's generated by the two element Boolean algebra \((B,\wedge,\vee)\) with \(a+b=a\vee b\) and \(a\dot-b=a\wedge b'\) consists of all Boolean algebras and all prime ideals of arbitrary Boolean algebras. 3) \(V({\mathbb{N}})\) and V(2) are the only equational complete varieties of CMM's.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 3875544
zbMATH Keywords: commutative monoid with monus
zbMATH Keywords: positive cones
zbMATH Keywords: lattice ordered groups
zbMATH Keywords: Boolean algebras
zbMATH Keywords: prime ideals
zbMATH Keywords: equational complete varieties
reviewed by: H. Mitsch
MaRDI profile type: Publication
cites work: Komplementäre Halbgruppen. Ein Beitrag zur instruktiven Idealtheorie kommutativer Halbgruppen
cites work: A representation of relatively complemented distributive lattices
cites work: Free lattice-ordered Abelian groups. II
full work available at URL: https://doi.org/10.1007/bf01182254
OpenAlex ID: W2069545812
description: scientific article; zbMATH DE number 3875544