Subdirectly irreducible semilattices with an automorphism

In other words, SA is the variety of semilattices with one automorphism (which is, as well as its inverse, considered as an additional fundamental operation). The aim of this paper is to find all subdirectly irreducible algebras in SA . A universal algebra A is said to be subdirectly irreducible (shortly, an SI algebra) if it contains more than one element and among all the nontrivial congruences of A there exists a least one; nontrivial means different from idA = {(a, a); a ∈ A} . The largest example of an SI algebra in SA is the algebra P(Z) defined as follows. Its underlying set is the set of all subsets of Z (where Z denotes the set of integers); the operations are defined by