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A concept of variety for regular biordered sets

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Abstract

We define a bivariety of regular biordered sets to be a nonempty class of regular biordered sets which is closed under taking direct products, regular bimorphic images and relatively regular biordered subsets. It is then shown that there is a complete lattice morphism mapping the complete lattice of all e-varieties of regular semigroups onto the complete lattice of all bivarieties of regular biordered sets; as a corollary we prove that there is a complete lattice morphism mapping the complete lattice of all e-varieties of E-solid regular semigroups onto the complete lattice of all varieties of solid binary algebras. Examples of bivarieties include the class of all solid regular biordered sets and the class of all local semilattices. For each setX with at least two elements, we show that a bivariety contains a free object onX if and only if it consists entirely of solid regular biordered sets or it consists entirely of local semilattices.

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Authors and Affiliations

  1. Department of Mathematics, Monash University, 3168, Clayton, Victoria, Australia

    Raymond Broeksteeg

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  1. Raymond Broeksteeg
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Communicated by T. E. Hall

The author gratefully acknowledges the financial support of an Australian Postgraduate Research Award.

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Broeksteeg, R. A concept of variety for regular biordered sets. Semigroup Forum 49, 335–348 (1994). https://doi.org/10.1007/BF02573495

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  • DOI: https://doi.org/10.1007/BF02573495

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