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On Equations for Union-Free Regular Languages

In this paper we consider the variety UF generated by all algebras of binary relations equipped with the operations of composition, reflexive-transitive closure, and the empty set and the identity relation as constants. This variety coincides with the variety generated by the union-free reducts of Kleene algebras of languages and its free objects are formed by union-free regular languages, that is, regular languages represented by regular expressions having no occurrence of +. We show that the variety UF is not finitely based. The situation does not change if we consider the variety UF? generated by the above algebras of binary relations equipped with the conversion operation.