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.

Mathematics Subject Classification (2020):

• Primary: 06A12
• 06A15
• 18A20
• Secondary: 06B35
• 18A30
• 18B35
• 54E15

Key words:

• Closure operator
• regular closure
• join semilattice
• epimorphism problem
• regular monomorphism
• generalized quasi-metric semilattice
• zigzag theorem