On equationally compact semilattices

Although equationally compact semilattices have been completely characterized [4], the question of J. Mycielski "Is every equationally compact semilattice the retract of a compact topological semilattice?" has heretofore remained unanswered. The main purpose of the present paper is to provide an affirmative answer to this question. Further, a new notion of "algebraic" compactness is introduced which among all semilattices singles out exactly those in which every chain is finite. Such semilattices are in turn compact topological ones in view of the more general result that the class of compact topological semilattices includes all join-complete semilattices in which every chain has a least element. Throughout this paper the term "semilattice" shall mean "join semilattice". The results presented here form a part of the author's doctoral thesis. For inspiration and guidance during the course of this investigation the author expresses gratitude to G.H. Wenzel