Expansions of finite algebras and their congruence lattices

In this paper we present a novel approach to the construction of new finite algebras and describe the congruence lattices of these algebras. Given a finite algebra 〈B, . . .〉, let B1, B2, . . . , BK be sets which intersect B at specific points. We construct an overalgebra 〈A,FA〉, by which we mean an expansion of 〈B, . . .〉 with universe A := B ∪ B1 ∪ · · · ∪ BK , and a certain set FA of unary operations which include idempotent mappings e and ei satisfying e(A) = B and ei(A) = Bi. We explore a number of such constructions and prove results about the shape of the new congruence lattices Con 〈A,FA〉 that result. Thus, descriptions of some new classes of finitely representable lattices is one contribution of this paper. Another, perhaps more significant contribution is the announcement of a novel approach to the discovery of new classes of representable lattices, the full potential of which we have only begun to explore.