Subdirectly irreducible semilattices with endomorphism

Abstract

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;∧,k)(S;∧,k) in which (S;∧)(S;∧) is a meet semilattice and kk is an endomorphism on SS characterised by the property k⩾idSk⩾idS. We particularly show that such an algebra is subdirectly irreducible if and only if it is a chain with one of the following forms ⋯<aj<aj−1<⋯<a0⋯<aj<aj−1<⋯<a0;0⋯<aj<aj−1<⋯<a00⋯<aj<aj−1<⋯<a0 in which k(aj)=aj−1k(aj)=aj−1 for j⩾1j⩾1, k(0)=0k(0)=0 and k(a0)=a0k(a0)=a0.