2003•Discussiones Mathematicae - General Algebra and ApplicationsOpen access

On lattice-ordered monoids

Milan Jasem

Abstract

In the paper lattice-ordered monoids and specially normal latticeordered monoids which are a generalization of dually residuated latticeordered 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 2 A and na ‚ 0 for some positive integer n imply a ‚ 0. A necessary and su‐cient 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 µ I, to be the direct product of the lattice-ordered group I and a lattice-ordered semigroup P with the least element 0.

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Key concepts: Mathematics, Lattice (music), Monoid, Invertible matrix, Semigroup, Bicyclic semigroup, Direct product, Combinatorics