Weak representations of interval algebras

This paper considers the algebra An of n-intervals, which coincides with Allen's algebra for n=2, and proves that An has a unique countable representation up to isomorphism for all n2 1.

Abstract

Ladkin and Maddux [LaMa87] showed how to interpret the calculus of time intervals defined by Allen [All83] in terms of representations of a particular relation algebra, and proved that this algebra has a unique countable representation up to isomorphism. In this paper, we consider the algebra An of n-intervals, which coincides with Allen's algebra for n=2, and prove that An has a unique countable representation up to isomorphism for all n2 1. We get this result, which implies that the first order theory of An is decidable, by introducing the notion of a weak representation of an interval algebra, and by giving a full classification of the connected weak representations of An. We also show how the topological properties of the set of atoms of An can be represented by a n-dimensional polytope.