Subdirectly irreducible semilattices with endomorphism

In this paper we initiate an investigation into the class of meet semilattices endowed with an endomorphism. A consideration of the subdirectly irreducible algebras leads to a description of a subclass of those algebras (S; Lambda, k) in which (S; Lambda) is a meet semilattice and k is an endomorphism on S characterised by the property k >= id(S). We particularly show that such an algebra is subdirectly irreducible if and only if it is a chain with one of the following forms: (1)...< a(j) < a(j-1) < ... < a(0); (2) 0 < ... < a(j) < a(j-1) < ... < a(0) in which k(a(j)) = a(j-1) for j >= 1, k(0) = 0 and k(a(0)) = a(0).