Relations between sets of time points and quasi-linear orderings

Wojciech Buszkowski

pp. 301-321

in: Jan Faye, UWE Scheffler, Max Urchs (eds), Perspectives on time, Berlin, Springer, 1997

Abstract

The mathematical contents of this paper are quite elementary. We consider binary relations R ⊂ P(U) × P(U), i.e. relations between subsets of a universe U. We prove that relations of that kind which satisfy certain additiveness conditions can be characterized as those which admit some simple first-order definition involving two unary predicates and a binary predicate, interpreted as a so-called quasi-linear ordering on U. Thus, this topic is obviously connected to the traditional concern of Jónsson and Tarski [7] with additive operators definable by means of binary relations, which is an algebraic origin for Kripke frames for modal logics [2]. Some recent generalization of the Jónsson and Tarski approach to n-ary operators definable by means of n +1-ary relations can be found in Dunn [3, 4]. Yet, our perspective is different from that of Jónsson, Tarski and Dunn, since we deal with relations instead of operators, which enables us to obtain stronger results in some respect.