Some properties of conjunctivity (subfitness) in generalized settings

Abstract

The property of subfitness used in point-free topology (roughly speaking) to replace the slightly stronger T1-separation, appeared (as  disjunctivity) already in the pioneering Wallman’s [16], then practically disappeared to reappear again (conjunctivity, subfitness), until it  was in the recent decades recognized as an utmost important condition playing a very special role. Recently, it was also observed that this property (or its dual) appeared independently in general poset setting (e.g. as separativity in connection with forcing). In a recent  paper [2], Delzell, Ighedo and Madden discussed it in the context of semilattices. In this article we discuss it on the background of the  systems of meet-sets (subsets closed under existing infima) in posets of various generality (semilattices, lattices, distributive lattices,  complete lattices) and present parallels of some localic (frame) facts, including a generalized variant of fitness.