Compactness of systems of equations in semigroups

Abstract

We considei systems u i = v i (i ∃ I) of equations in semigroups over finite sets of variables. A semigroup S is said to satisfy the compactness property (or CP, for short), if each system of equations has an equivalent finite subsystem. It is shown that all monoids in a variety V satisfy CP, if and only if the finitely generated monoids in V satisfy the maximal condition on congruences. We also show that if a finitely generated semigroup S satisfies CP, then S is necessarily hopfian and satisfies the chain condition on idempotents. Finally, we give three simple examples (the bicyclic monoid, the free monogenic inverse semigroup and the Baumslag-Solitar group) which do not satisfy CP, and show that the above necessary conditions are not sufficient.