Continuity of posets via Scott topology and sobrification

Abstract

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:(1) A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;(2) A poset is continuous iff its Scott topology is completely distributive;(3) A topological T 0 space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology; (4) A topological T 1 space is a discrete space iff its topology is completely distributive.These results generalize the relevant results obtained by J.D. Lawson for dcpos.

A note on continuities of the poset of Turing degrees

“…To go further, we have Definition 3.9. (see [7]) Let B and P be posets. If there is a map j : B → P satisfying (1) j preserves existing directed sups, (2) j : B → j(B) is an order isomorphism, (3) j(B) is a basis for P , then (B, j) is called an embedded basis for P .…”

Quasicontinuity of Posets via Scott Topology and Sobrification

“…It is known that the directed completion c(P) of a continuous poset P is isomorphic to the round ideal completion RI(P) on the abstract basis (P, ) and (c(P), σ (c(P))) is a concrete sobrification of (P, σ (P)) (see [9,13]). Thus directed completions of continuous posets can be viewed as concrete realizations of round ideal completions and sobrifications.…”

“…Since there are more and more demands to study posets which are not directed complete (see [10][11][12][13]), this paper manages to generalize the concept of quasicontinuity to the setting of general posets in terms of the Scott topology defined for all posets. We will see that quasicontinuous posets have many properties similar to that of quasicontinuous domains.…”

Scott convergence and fuzzy Scott topology on L-posets

“…Unfortunately, they are not fit for arbitrary partially ordered sets (posets), since the join of a directed subset is involved in the definition of Scott convergence, which may not exist in a poset. Regarding this, several alternative choices have been proposed to generalize the definition of Scott convergence in posets [1,[6][7][8][9][10][11], and the Scott topology related to Scott convergence has also been studied.…”