Non Axiomatisability of Positive Relation Algebras with Constants, via Graph Homomorphisms

Authors Amina Doumane, Damien Pous

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Amina Doumane
  • Université Lyon, CNRS, ENS de Lyon, UCB Lyon 1, LIP, Lyon, France
Damien Pous
  • Université Lyon, CNRS, ENS de Lyon, UCB Lyon 1, LIP, Lyon, France

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Amina Doumane and Damien Pous. Non Axiomatisability of Positive Relation Algebras with Constants, via Graph Homomorphisms. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We study the equational theories of composition and intersection on binary relations, with or without their associated neutral elements (identity and full relation). Without these constants, the equational theory coincides with that of semilattice-ordered semigroups. We show that the equational theory is no longer finitely based when adding one or the other constant, refuting a conjecture from the literature. Our proofs exploit a characterisation in terms of graphs and homomorphisms, which we show how to adapt in order to capture standard equational theories over the considered signatures.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Relation algebra
  • graph homomorphisms
  • (in)equational theories


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