Subdirectly irreducible semilattices with endomorphism

In this paper we initiate an investigation into the class of meet semilattices endowed with endomorphism. A consideration subdirectly irreducible algebras leads to a description subclass those (S;∧,k)(S;∧,k) in which (S;∧)(S;∧) is semilattice and kk endomorphism on SS characterised by property k⩾idSk⩾idS. We particularly show that such algebra if only it chain one following forms ⋯<aj<aj−1<⋯<a0⋯<aj<aj−1<⋯<a0;0⋯<aj<aj−1<⋯<a00⋯<aj<aj−1<⋯<a0 k(aj)=aj−1k(aj)=aj−1 for j⩾1j⩾1, k(0)=0k(0)=0 k(a0)=a0k(a0)=a0.