Continuity of posets via Scott topology and sobrification

Quantale-valued fuzzy Scott topology

The aim of this paper is to extend the truth value table oflattice-valued convergence spaces to a more general case andthen to use it to introduce and study the quantale-valued fuzzy Scotttopology in fuzzy domain theory. Let $(L,*,varepsilon)$ be acommutative unital quantale and let $otimes$ be a binary operationon $L$ which is distributive over nonempty subsets. The quadruple$(L,*,otimes,varep...

Meet-continuity on $L$-directed Complete Posets

In this paper, the definition of meet-continuity on $L$-directedcomplete posets (for short, $L$-dcpos) is introduced. As ageneralization of meet-continuity on crisp dcpos, meet-continuity on$L$-dcpos, based on the generalized Scott topology, ischaracterized. In particular, it is shown that every continuous$L$-dcpo is meet-continuous and $L$-continuous retracts ofmeet-continuous $L$-dcpos are al...

Formal Topology with Posets

Topology and Continuity

During the last 200 years mathematics has enjoyed a return of geometric techniques. For one who has not studied the subject at a higher level, “geometry” may be a word reminiscent of theorems about triangles and circles dating back to ancient Greece, possibly the topic of some middle and high school classes. Of course, there have been numerous developments in the subject dating from ancient tim...

The Scott Topology . Part II

The following propositions are true: (1) Let X be a set and F be a finite family of subsets of X. Then there exists a finite family G of subsets of X such that G ⊆ F and ⋃ G = ⋃ F and for every subset g of X such that g ∈ G holds g 6⊆ ⋃ (G \ {g}). (2) Let S be a 1-sorted structure and X be a subset of the carrier of S. Then −X = the carrier of S if and only if X is empty. (3) Let R be an antisy...