s2-Quasicontinuous posets

Abstract

In this paper, we consider a common generalization of both s 2 -continuous posets and quasicontinuous domains, and we introduce new concepts of way below relations and s 2 -quasicontinuous posets. The main results are: (1) The way below relation on an s 2 -quasicontinuous poset has the interpolation property; (2) The λ 2 -topology on an s 2 -quasicontinuous poset is completely regular; (3) A poset is s 2 -continuous iff it is meet s 2 -continuous and s 2 -quasicontinuous.

Convergence in $$s_{2}$$ s 2 -quasicontinuous posets

“…(Zhang and Xu 2015 ) Let P be a poset and G , , we say that G is way below H and write if for all directed sets , implies . We write for and for .…”

“…(Zhang and Xu 2015 ) Let P be an - quasicontinuous poset and , . If , then there exists a finite set F such that .…”

“…Since many models may not be dcpos, an important direction in the study of continuous domains is to extend the theory of continuous domains to that of posets as much as possible (see Huang et al. 2009 ; Lawson and Xu 2004 ; Mislove 1999 ; Markowsky 1981 ; Mao and Xu 2006 ; Venugopalan 1990 ; Zhang 1993 ; Zhang and Xu 2015 ). It has turned out to be very fruitful for many categorical and topological developments generalizing the theory of continuous domains, but it is still rather restrictive, taking into consideration only the case of existing a join.…”

“…The notion of -continuity admits to generalize most important characterizations of continuity from dcpos to arbitrary posets and has the advantage that not even the existence of directed joins has to be required. As a generalization of -continuity, the concept of -quasicontinuity was introduced by Zhang and Xu ( 2015 ), their basic idea is to generalize the way below relation between the points to the case of sets. It was proved that -quasicontinuous posets equipped with the weak Scott topologies are precisely the hypercontinuous lattices.…”

θ-continuity and Dθ-completion of posets

“…To avoid the requirement of the existence of directed joins, Erné introduced s 2 -continuous posets, which allow to generalize important characterizations of continuity from complete lattices to arbitrary posets (Erné 1981). In the manner of Erné, Zhang and Xu came up with a new way below relation and used it to define s 2 -quasicontinuous posets as a common generalization of both s 2 -continuous posets and quasicontinuous domains (Zhang and Xu 2015). Recall that a complete lattice L is called meet continuous if it satisfies the distributive law that binary meets distribute over directed joins.…”