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Motivated by a problem posed in [10], we investigate the closure operators of the category  SLatt  of join semilattices and its subcategory  SLattO  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  SLattO  are surjective. We complement the paper with two different proofs of this result using either generators or Isbell’s zigzag theorem.

Epimorphisms and closure operators of categories of semilattices