Partially ordered sets with projections and their topology

Abstract

The family F gives rise to a uniformity on D, which we call the F-uniformity. The uniform topology is the F-topology. We characterize all uniformities on a poset arising as F-uniformities and investigate basic properties of F-uniformity and F-topology. As we have already emphasized, the mappings in F will be used for approximation purposes. We therefore say that an F-poset (D, ≤, F) is approximating if sup f ∈F f (d) = d for all d ∈ D. It turns out that this order-theoretic condition is equivalent to saying that the poset is a partially ordered space in the F-topology. We shall mainly focus on approximating F-posets. Then a close relationship between the partial order and the F-topology can be established. We are interested in the question when suprema and infima of subsets A ⊆ D exist and, moreover, when they are accumulation points of A with respect to the F-topology. Here continuous domains come into play. We show that each monotone net in D converges in the F-topology if and only if (D, ≤) is a continuous domain such that f (d) is "essentially" below d for all f ∈ F and all d ∈ D. In addition, a similar result holds concerning the convergence of monotone nets which are bounded. We give a domain-theoretic characterization of approximating F-posets that are compact in their F-topology. In this case, the F-topology coincides with the Lawson topology of the poset. Moreover, provided that they have a least element, these F-posets arise exactly from FS-domains. Part of these results can also be found in [33].Chapter 3 introduces a special sort of F-posets involving projections. Therefore, we first collect some basic facts on projections. After that, we define a partially ordered set with projections (or pop for short) to be an F-poset (D, ≤, P) where P consists of projections only. We investigate the pop uniformity and the pop topology, i.e. the Funiformity and the F-topology of a pop, and give a list of examples. For instance, Flagg and Kopperman's [19] closed ball model carries such a pop structure. Moreover, we show how several domains of traces can be made into pop's. This enables us to apply our results in order to obtain a uniform proof for topological properties shared by all these trace models.Coming back to the general theory, we show that approximating pop's are complete in their pop uniformity if and only if they can be represented as an inverse limit built up by the images of their projections. We resume the problem when the supremum or the infimum of a subset A exists in the closure of A. This leads to algebraic domains. Using the results of Chapter 2, we prove that each monotone net of an approximating pop (D, ≤, P) converges in the pop topology if and only if (D, ≤) is an algebraic domain with its compact elements being exactly the images of all projections of P. Furthermore, we raise the question under which (order-theoretic) conditions an algebraic domain (D, ≤) admits such a set P. We then call (D, ≤) a P-domain. Here posets satisfying the ascending chain condition (ACC) come into play. As we s...

Directed*-topology and Scott*-topology on Transitive Binary Relational Sets

“…This work is devoted to introduce and study the continuity and algebraicness properties of TRS. Our results extended the results in posets and in domains [1][2][3]13]. The concepts of upper bound (for short ub), lower bound (for short lb), least upper bound (for short), gretest lower bound (for short) in any poset are clear also, some concepts in mathematical logics my building some times needs these facts [14].…”

“…In domain and poset [1][2][3], Scott-topologies were defined. Abramsky and Jung [4] introduced the concepts of continuous directed complete posets (continuous domain) and algebraic domains.…”