Epimorphisms and closure operators of categories of semilattices

Abstract

Motivated by a problem posed in [10], we investigate the closure operators of the category SLatt of join semilattices and its subcategory SLatt O of join semilattices with bottom element. In particular, we show that there are only finitely many closure operators of both categories, and provide a complete classification. We use this result to deduce the known fact that epimorphisms of SLatt and SLatt O are surjective. We complement the paper with two different proofs of this result using either generators or Isbell’s zigzag theorem.