REPRESENTING SUBDIRECT PRODUCT MONOIDS BY GRAPHS

Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M. In a previous paper, we give a construction G(M) for a graph with prescribed endomorphism monoid M known as a [Formula: see text]-graph. Using this construction, we derived bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. In this paper, we generalize the [Formula: see text]-graph construction and derive several new bounds for monoid classes not handled by our first paper. Among these are the so called "strong semilattices of C-semigroups" where C is one of the following: Groups, Abelian Groups, Rectangular Groups, and completely simple semigroups.