Decidability of Classes of Finite Algebras with a Distinguished Subset Closed Under a Discriminator Clone

Let C be a clone of operations on a two element set. In [2] we have proved that the first order theory of finite Boolean algebras with distinguish subset closed under C is decidable if and only if C is a clone containing the ternary discriminator function. Decidability of the theories referred to in the abstract is a consequence of the fact, that finite discriminator subsets of products of algebras have a ’nice’ ordered structure, especially in case of Boolean algebras [3].