Undecidability of satisfiability in the algebra of finite binary relations with union, composition, and difference

We consider expressions built up from binary relation names using the operators union, composition, and set difference. We show that it is undecidable to test whether a given such expression $e$ is finitely satisfiable, i.e., whether there exist finite binary relations that can be substituted for the relation names so that $e$ evaluates to a nonempty result. This result already holds in restriction to expressions that mention just a single relation name, and where the difference operator can be nested at most once.

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