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On polars and direct decompositions of lattice ordered monoids. (English) Zbl 1078.06008

Chajda, I. (ed.) et al., Proceedings of the 68th workshop on general algebra “68. Arbeitstagung Allgemeine Algebra”, Dresden, Germany, June 10–13, 2004 and of the summer school 2004 on general algebra and ordered sets, Malá Morávka, Czech Republic, September 5–11, 2004. Klagenfurt: Verlag Johannes Heyn (ISBN 3-7084-0163-8/pbk). Contributions to General Algebra 16, 115-131 (2005).
The goal of the present paper is to investigate some types of orthogonality and polars in lattice-ordered monoids (\(l\)-monoids). It is shown that many properties and results concerning these concepts in the cases of lattice-ordered groups (\(l\)-groups) and of dually residuated lattice-ordered monoids (\(DRl\)-monoids) also hold for \(l\)-monoids. One of the main results states that the set consisting of all polars in an \(l \)-monoid partially ordered by inclusion is a complete complemented lattice (see Theorem 14). Another important theorem of the paper is Theorem 17. Here, considering a commutative \(l\)-monoid \(A\) and assuming that the set \(I\) of all invertible elements in \(A\) is a convex subset of \(A\), the author indicates a necessary and sufficient condition for \(A\) in order to be the direct product of the lattice-ordered group \(I\) and the polar \(I^{\perp }\).
For the entire collection see [Zbl 1067.08002].

MSC:

06F05 Ordered semigroups and monoids
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