Varieties of lattice-ordered algebras
type: Item
instance of: scholarly article
title: Varieties of lattice-ordered algebras
zbMATH Open document ID: 0535.06011
DOI: 10.1007/BF01194545
author: N. E. Zubov
published in: Algebra Universalis
publication date: 1983-01-01T00:00:00Z
review text: The variety of lattice-ordered groups generated by fully ordered groups is axiomatised either by (i) \((x\vee y)^ 2=x^ 2\vee y^ 2\) or (ii) \((x\vee 1)\wedge y^{-1}(x^{-1}\vee 1)y=1\). The variety of lattice- ordered loops generated by fully ordered loops is axiomatised by: (a) \((x/z\vee 1)\wedge y\backslash((z/x\vee 1)y)=1\) (b) \((x/z\vee 1)\wedge(((z/x\vee 1)y)t)/yt=1\) (c) \((x/z\vee 1)\wedge(ty\backslash(t(y(z/x\vee 1))))=1.\) This shows that the associative law can largely be dispensed with - an interesting result. The other important theorem in this paper is that every lattice-ordered loop is distributive. The authors provide a start to the study of lattice-ordered loops and challenge others to continue it.
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 3849297
zbMATH Keywords: variety of lattice-ordered groups
zbMATH Keywords: variety of lattice-ordered loops
zbMATH Keywords: fully ordered loops
reviewed by: A. M. W. Glass
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3849297