On minimum proper essential extensions in a category

Abstract

In a concrete category A satisfying certain natural conditions, we investigate minimum proper essential extenstions (A-mpee’s) and how this relates to atoms in the complete lattice hoA of hull operators on A (if A # is the category with objects the A-objects but morphisms just the essential extensions in A, then hoA consists exactly of the reflections in A # ). A further hypothesis inserts itself: We call an A-mpee µ strong just in case the domain and codomain of µ are not isomorphic; [S] is the assumption that every A-mpee is strong. We show (1) if µ is strong, then an atom of hoA is determined by µ, and (2) if [S], then every id ≠ h ∈ hoA lies above an atom and every atom is determined by an A-mpee µ. Examples are presented in various categories of lattice-ordered groups.