Notes on Exact Meets and Joins

Abstract

An exact meet in a lattice is a special type of infimum characterized by, inter alia, distributing over finite joins. In frames, the requirement that a meet is preserved by all frame homomorphisms makes for a slightly stronger property. In this paper these concepts are studied systematically, starting with general lattices and proceeding through general frames to spatial ones, and finally to an important phenomenon in Scott topologies.

A pointfree theory of Pervin spaces

“…We say that a bispace (X, τ + , τ − ) is zero-dimensional if every element in τ + is a union of positive clopens, and every element in τ − is a union of negative clopens. 1 A biframe is a triple L = (L, L + , L − ) such that all three components are frames, together with subframe inclusions L + ⊆ L and L − ⊆ L, and such that every element of L is a join of finite meets of elements of L + ∪L − . The frame L is called the main component of the biframe, while L + and L − are, respectively, the positive and negative components.…”

“…Let now L be a frame and P ⊆ L. The meet P is strongly exact if the corresponding intersection of open sublocales (=open pointfree subspaces) is open, or equivalently, if the congruence s∈P ∆ s is open (cf. [1,Section 4.5]). Note that, if ∆ a = s∈P ∆ s for some P ⊆ L, then the meet P is, by definition, strongly exact, and we necessarily have a = P .…”

Pervin Spaces and Frith Frames: Bitopological Aspects and Completion

“…Let now L be a frame and P ⊆ L. The meet P is strongly exact if the corresponding intersection of open sublocales (=open pointfree subspaces) is open, or equivalently, if the congruence s∈P s is open (cf. [1,Section 4.5]). Note that, if a = s∈P s for some P ⊆ L, then the meet P is, by definition, strongly exact, and we necessarily have a = P. Given a Frith frame (L, S), we denote by [ S ] se the set of elements of L that may be written as a strongly exact meet of elements of S. Again, [ S ] se can be thought of as the closure of S under strongly exact meets.…”