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Discussiones Mathematicae General Algebra and Applications 23(2) (2003) 101-114
DOI: https://doi.org/10.7151/dmgaa.1066

ON LATTICE-ORDERED MONOIDS

Milan Jasem

Department of Mathematics, Faculty of Chemical Technology
Slovak Technical University
Radlinského 9, 812 37 Bratislava, Slovak Republic
e-mail: milian.jasem@stuba.sk

Abstract

In the paper lattice-ordered monoids and specially normal lattice-ordered monoids which are a generalization of dually residuated lattice-ordered semigroups are investigated. Normal lattice-ordered monoids are metricless normal lattice-ordered autometrized algebras. It is proved that in any lattice-ordered monoid A, a ∈ A and na > 0 for some positive integer n imply a > 0. A necessary and sufficient condition is found for a lattice-ordered monoid A, such that the set I of all invertible elements of A is a convex subset of A and A^- ⊆ I, to be the direct product of the lattice-ordered group I and a lattice-ordered semigroup P with the least element 0.

Keywords: lattice-ordered monoid, normal lattice-ordered monoid, dually residuated lattice-ordered semigroup, direct decomposition, polar.

2000 Mathematics Subject Classification: 06F05.

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Received 26 November 2002
Revised 16 May 2003
Revised 20 November 2003