Elzbieta Hajnicz

Defining intervals by means of point skeletons: generalization of pairs of points

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Abstract

This report is a continuation of our earlier work on time intervals defined in the non-linear point time framework. It turns out that pairs of points, which are a very convenient way of representing intervals, in non-linear time form a class of intervals too small to be considered as a satisfactory representation. In this report, a notion of {\it point skeletons} is introduced, being a generalization of a notion of pairs of points adequate for non-linear time enabling us to represent a substantially larger classes of intervals than pairs of points.

Point skeletons are organized by a precedence relation $\prec$ and the overlapping relation $\subseteq$. Three axiomatizations are presented, for three different precedence relations which could be naturally defined for point skeletons (being counterparts of precedence relation defined for intervals represented as sets of points). Each axiomatization is proven to be complete.

Key words: non-linear time structures, time points, time intervals