Homomorphic Images of Finite Subdirectly Irreducible Unary Algebras

We prove that a finite unary algebra with at least two operation symbols is a homomorphic image of a (finite) subdirectly irreducible algebra if and only if the intersection of all its subalgebras which have at least two elements is nonempty. We are concerned with the following question: Which algebras are homomorphic images of subdirectly irreducible algebras? A necessary condition, discovered in [3], [4] and [5], for an algebra A with at least one at least binary operation to be a homomorphic image of some subdirectly algebra, is that the intersection of all ideals of A is nonempty. (By an ideal of A we mean a nonempty subset I such that f(a1, . . . , an) ∈ I whenever f is a fundamental operation and a1, . . . , an are elements of A with ai ∈ I for at least one i.) It was proved in [2] and independently in [7] that the condition is also sufficient. In fact, it was proved in those two papers that any algebra A with at least one at least binary operation and with a smallest ideal is isomorphic to a factor of a subdirectly irreducible algebra B through its monolith, and the construction is such that if A is finite then also B is finite. The case that remains is that of algebras with only unary and nullary operations. (However, note that nullary operations play no role in investigation of congruences.) If there is just one unary operation, the characterization is simple; see e.g. [8] or [2]. For two or more unary operations, the situation is more complicated. In the present paper we are going to characterize finite algebras with at least two unary operations that are homomorphic images of a subdirectly irreducible algebra. We leave it as an open problem to do the same for infinite algebras. An example found in [2] suggests that it will be probably much harder to characterize such unary algebras that are isomorphic to a subdirectly irreducible algebra through its monolith. 1991 Mathematics Subject Classification. 08A60, 08B26.