Topological representations of distributive hypercontinuous lattices

Abstract

The concept of locally strong compactness on domains is generalized to general topological spaces. It is proved that for each distributive hypercontinuous lattice L, the space SpecL of nonunit prime elements endowed with the hull-kernel topology is locally strongly compact, and for each locally strongly compact space X, the complete lattice of all open sets O(X) is distributive hypercontinuous. For the case of distributive hyperalgebraic lattices, the similar result is given. For a sober space X, it is shown that there is an order reversing isomorphism between the set of upper-open filters of the lattice O(X) of open subsets of X and the set of strongly compact saturated subsets of X, which is analogous to the well-known Hofmann-Mislove Theorem.

The dual of a generalized completely distributive lattice is a hypercontinuous lattice

“…Since x ≺ y in L if and only if there exists a finite subset F ⊆ L such that y ∈ ↓F and L \ ↓F ⊆ ↑x, one can easily get the following intrinsic characterization of hypercontinuous lattices (see, e.g., [4,7,8]). …”

“…(1) ⇒ (2) : Since L is GCD iff L op is hypercontinuous by Theorem 2.5, (2) follows directly from Theorem 2.1 of [8].…”

Dually normal relations on sets

“…Definition 3.4 (see [6]) Let ρ be a binary relation on a set X. Define a binary relation ρ (<ω) on X (<ω) , called the finite extension of ρ, by…”

Split Hausdorff internal topologies on posets

“…Further criteria for regularity were given by Markowsky [19] and Schein [20] (see also [21] and [22]). Motivated by the fundamental works relative Zareckiǐ on regular relations, Xu and Liu [23] introduced the concepts of nitely regular relations and generalized nitely regular relations, respectively. It is proved that a relation ρ is generalized nitely regular if and only if the interval topology on (Φρ(X), ⊆) is T .…”

“…Therefore, we introduce the concepts of the split T interval topology on posets and weak generalized nitely regular relations. Meanwhile, in order to characterize split T interval topology of posets by a order structure, like the equivalence of the T interval topology and quasi-hypercontinuous lattices in [23], we give the notion of a weak quasi-hypercontinuous poset. It is proved that when a binary relation ρ : X Y satis es property M, ρ is weak generalized nitely regular if and only if (φρ(X, Y), ⊆) is a weak quasihypercontinuous poset if and only if the interval topology on (φρ(X, Y), ⊆) is split T , where φρ(X, Y) = {ρ(x) : x ∈ X}.…”