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Notes on equational theories of relations

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Abstract

We describe explicitly the free algebras in the equational class generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure and neutral elements 0 (empty relation) and 1 (identity relation). We show the corresponding equational theory is decidable by reducing the problem to a question about regular sets. Similar results are given for two related equational theories.

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Authors and Affiliations

  1. Computer Science Department, A. József University, Aradi V. Tere 1, 6720, Szeged, Hungary

    S. L. Bloom, Z. Ésik & Gh. Stefanescu

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  1. S. L. Bloom
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  2. Z. Ésik
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  3. Gh. Stefanescu
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Additional information

Partially supported by a joint grant from the NSF and the Hungarian Academy of Sciences.

Partially supported by a grant from the Hungarian National Foundation for Scientific Research and a joint grant from the NSF and the Hungarian Academy of Sciences.

v-semirings of 1-closed regular sets. On the basis of this characterization, we conjectured that a set of equational axioms for the variety RELv consists of equational axioms for the variety Lv and the equation (10). Recently, this conjecture has been proved in [6].

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Bloom, S.L., Ésik, Z. & Stefanescu, G. Notes on equational theories of relations. Algebra Universalis 33, 98–126 (1995). https://doi.org/10.1007/BF01190768

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