Reconstructing an Open Order from Its Closure, with Applications to Space-Time Physics and to Logic

Abstract

Abstract. In his logical papers, Leo Esakia studied corresponding ordered topological spaces and order-preserving mappings. Similar spaces and mappings appear in many other application areas such the analysis of causality in space-time. It is known that under reasonable conditions, both the topology and the original order relation can be uniquely reconstructed if we know the "interior" ≺ of the order relation. It is also known that in some cases, we can uniquely reconstruct ≺ (and hence, topology) from . In this paper, we show that, in general, under reasonable conditions, the open order ≺ (and hence, the corresponding topology) can be uniquely determined from its closure .Keywords: ordered topological space, order-preserving mappings, open and closed orders, space-time geometry, logic Formulation of the ProblemOrder-preserving mappings of topological spaces in logic and in physics: general reminder. Many interesting mathematical results appear when we are able to find connection between two seemingly different areas of mathematics -and thus, use known results and techniques from one area to study techniques from another area. In particular, for Heyting algebras -models of intuitionistic logics -many results originated from a relation between Heyting algebras and a special class of (partially) ordered topological spaces called Esakia spaces, a relation that was discovered and actively explored by Leo Esakia in [10,11]. In his research, L. Esakia paid special attention to studying order-preserving maps between the corresponding partially ordered spaces.Esakia's work was not the first application of ordered topological spaces and order-preserving mappings: such spaces and mappings also naturally appear in space-time physics and in other areas of logic.In physics, a natural ordering relation is the causality relation between events, when a b means that an event a can influence the event b.

Causal and Topological Aspects in Special and General Theory of Relativity

“…In the light of this definition, we shall observe in section 3 that causal cones and Kcausal cones fall in this category since causal relation < and K-causal relation ≺ are reflexive and transitive. In the literature, ( see for example [26,27,28]), cone preserving mappings are defined as follows: Let A = (A, R) and B = (B, S) be quasi-ordered sets. A mapping h :…”

“…In the recent paper, Zapata and Kreinovich [28] call chronological order as open order and causal order as closed order and prove that under reasonable assumptions, one can uniquely reconstruct an open order if one knows the corresponding closed order. For special theory of relativity, this part is true and hence every one-one transformation preserving a closed order preserves open order and topology.…”

Causal cones, cone preserving transformations and causal structure in special and general relativity

“…In the literature, ( see for example [25,26,27]), cone preserving mappings are defined as follows: Let A = (A, R) and B = (B, S) be quasi-ordered sets. A mapping h : A → B is called cone preserving if h(U R (a)) = U S (h(a)) for each a ∈ A.…”

“…In the recent paper, Zapata and Kreinovich [27] We now introduce the concept of K-causality and give causal properties of space-times in the light of this concept. For more details we refer the reader to [8], [10,11] and [30,31].…”