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 SLatt0 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 SLatt0 are surjective. We complement the paper with two different proofs of this result using either generators or Isbell’s zigzag theorem.