The equational theory of Kleene lattices

Languages and families of binary relations are standard interpretations of Kleene algebras. It is known that the equational theories of these interpretations coincide and that the free Kleene algebra is representable both as a relation and as a language algebra. We investigate the identities valid in these interpretations when we expand the signature of Kleene algebras with the meet operation. In both cases, meet is interpreted as intersection. We prove that in this case, there are more identities valid in language algebras than in relation algebras (exactly three more in some sense), and representability of the free algebra holds for the relational interpretation but fails for the language interpretation. However, if we exclude the identity constant from the algebras when we add meet, then the equational theories of the relational and language interpretations remain the same, and the free algebra is representable as a language algebra, too. The moral is that only the identity constant behaves differently in the language and the relational interpretations, and only meet makes this visible.