A mereotopology based on sequent algebras. Issue 3 (2nd October 2017) - Vakarelov, Dimiter
Abstract: Mereotopology is an extension of mereology with some relations of topological nature like contact. An algebraic counterpart of mereotopology is the notion of contact algebra which is a Boolean algebra whose elements are considered to denote spatial regions, extended with a binary relation of contact between regions. Although the language of contact algebra is quite expressive to define many useful mereological relations (part-of, overlap, underlap) and mereotopological relations (external contact, tangential part-of, non-tangential part-of, self-connectedness), there are, however, some interesting mereotopological relations which are not definable in it. Such are, for instance, the relation of n-ary contact, internal connectedness and some others. To overcome this disadvantage, we introduce a generalisation of contact algebra, replacing the contact with a binary relation between finite sets of regions and a region, satisfying some formal properties of Tarski consequence relation. The obtained system is called sequent algebra, considered as an algebraic counterpart of a new mereotopology. We develop the topological representation theory for sequent algebras showing in this way certain correspondence between point-free and point-based models of space. As a by-product, we show how one logical relation in nature notion, Tarski consequence relation, may have also certain spatial (mereotopological) meaning.