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Representations of distributive lattice-ordered semigroups with binary relations

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Abstract

LetR(∩, ∪, ¦) denote the class of all algebras isomorphic to ones whose elements are binary relations and whose operations are union, intersection, and relation composition (or relative product) of relations. We prove thatR(∪, ∩, ¦) is not a variety and is not finitely axiomatizable. LetDLOS denote the class of all structures (A, ∨, ∧, ∘) where (A, ∨, ∧) is a distributive lattice, (A, ∘) is a semigroup and ∘ is additive w.r.t. ∨. We prove thatDLOS is the variety generated byR(∪, ∩, ¦), and moreover, if (A, ∨, ∧, ∘) ∈DLOS then it is representable whenever we disregard one of its operations.

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Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary

    H. Andréka

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  1. H. Andréka
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Research supported by Hungarian National Foundation for Scientific Research grant No. 1810.

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Andréka, H. Representations of distributive lattice-ordered semigroups with binary relations. Algebra Universalis 28, 12–25 (1991). https://doi.org/10.1007/BF01190407

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  • DOI: https://doi.org/10.1007/BF01190407

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