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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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References

  1. Andréka, H.,On the representation problem of distributive semilattice-ordered semigroups. Preprint (1988 Sept.), Budapest. Abstracted in: Abstracts of AMS Vol 10, No 2 (March 1989),*89T-08-21, p. 174.

    Google Scholar 

  2. Bredihin, D. A. andSchein, B. M.,Representations of ordered semigroups and lattices by binary relations. Colloq. Math.39 (1978), 1–12.

    Google Scholar 

  3. Schein, B. M., Personal communication (1988).

  4. Szabó, A., Personal communication (1989).

  5. Maddux, R.,Topics in relation algebras. Ph.D. Thesis, University of California, Berkeley, 1978.

    Google Scholar 

  6. Maddux, R.,Some nonrepresentable relation algebras. Notices. Amer. Math. Soc.23 (1976), A-431, A-557.

    Google Scholar 

  7. Maddux, R.,Non-finite-axiomatizability results for cylindric and relation algebras. J. of Symb. Logic, Vol. 54, No 3 (1989), 951–974.

    Google Scholar 

  8. Jónsson, B. andTarski, A.,Boolean algebras with operators. Part II. Amer. J. Math.74 (1952), 127–162.

    Google Scholar 

  9. Németi, I., Algebraizations of quantifier logics, an introductory overview. Studia Logica, to appear.

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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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