Compactness of Systems of Equations on Completely Regular Semigroups

A semigroup S is said to have the compactness property, or CP for short, if each system of equations over a nite set of variables has an equivalent nite subsystem, that is, having exactly the same solutions in S. We prove that a completely 0-simple semigroup S satis es CP if and only if the group G in a Rees matrix representation S = M0(I; G; ;P) satis es this property. Further, a variety of completely regular semigroups in which the nitely generated members have nite matrices is shown to satisfy CP if and only if the corresponding variety of groups satis es CP. It is then shown that the varieties of bands of groups satisfy the condition on nite matrices.