Abstract

Abstract In this paper, the concepts of weak quasi-hypercontinuous posets and weak generalized finitely regular relations are introduced. The main results are: (1) when a binary relation ρ : X ⇀ Y satisfies a certain condition, ρ is weak generalized finitely regular if and only if (φρ(X, Y), ⊆) is a weak quasi-hypercontinuous poset if and only if the interval topology on (φρ(X, Y), ⊆) is split T2; (2) the relation ≰ on a poset P is weak generalized finitely regular if and only if P is a weak quasi-hypercontinuous poset if and only if the interval topology on P is split T2.

Highlights

  • In domain theory, the interval topology and the Lawson topology are two important "two-sided" topologies on posets

  • The main results are: (1) when a binary relation ρ : X Y satis es a certain condition, ρ is weak generalized nitely regular if and only if (φρ(X, Y), ⊆) is a weak quasi-hypercontinuous poset if and only if the interval topology on (φρ(X, Y), ⊆) is split T ; (2) the relation ≰ on a poset P is weak generalized nitely regular if and only if P is a weak quasi-hypercontinuous poset if and only if the interval topology on P is split T

  • A basic problem is: When do the interval topology and the Lawson topology have T properties? In [5], Gierz and Lawson have discussed this problem for the Lawson topology, and proved that a complete lattice is a quasicontinuous lattice if and only if the Lawson topology is T

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Summary

Introduction

The interval topology and the Lawson topology are two important "two-sided" topologies on posets. A basic problem (see [1,2,3,4,5]) is: When do the interval topology and the Lawson topology have T properties? Erné [1] obtained several equivalent characterizations about T properties of the interval topology on posets. For a complete lattice L, Gierz and Lawson [5] proved that the interval topology on L is T if and only if L is a generalized bicontinuous lattice. The regularity of binary relations was rst characterized by Zareckii [18]. It is proved that a relation ρ is generalized nitely regular if and only if the interval topology on (Φρ(X), ⊆) is T. In complete lattices, this condition turns out to be equivalent both to the T interval topology and to the quasi-hypercontinuous lattices

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Aδ on
Shuzhen Luo and Xiaoquan Xu
Weak generalized nitely regular relations

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