Compactness of Systems of Equations in Semigroups

Abstract

We consider systems ui=vi(i∈I) of equations in semigroups over finite sets of variables. A semigroup (or a monoid) S is said to satisfy the compactness property (CP, for short), if each system of equations has an equivalent finite subsystem. We prove that all monoids in a variety [Formula: see text] satisfy CP if and only if the finitely generated monoids in [Formula: see text] satisfy the maximal condition on congruences. Consequently, all commutative monoids (and semigroups) satisfy CP. Also, if a finitely generated semigroup S satisfies CP, then S is necessarily hopfian and satisfies the chain condition on idempotents. It follows that the free inverse semigroups do not satisfy CP. Finally, we give two simple examples (the bicyclic monoid and the Baumslag-Solitar group) which do not satisfy CP, and show that the necessary conditions above are not sufficient.