The Relations of Supremum and Mereological Sum in Partially Ordered Sets

Abstract

This paper is devoted to mutual relationship between the relations of mereological sum and least upper bound (supremum) in partially ordered sets. We are mainly interested in the following problem: under what conditions (axioms) put on mereological sum the both relations coincide? The mutual relation between the relations in question is, among others, the object of scientific scrutiny in Pietruszczak (Metamereologia (in Polish). Nicolaus Copernicus University Press, Torun, 2000) and Pietruszczak (Logic Logical Philos 14(2), 211–234, 2005). There it is shown that in the so called classical mereology, which is the contemporary version of Stanislaw Leśniewski’s original system, mereological sum coincides with supremum relation. The first one to prove this property, in original language of Leśniewski’s mereology, was Alfred Tarski (see Leśniewski (Przeglad Filozoficzny, XXXIII, 77–105, 1930), p. 87). The paper starts with some refresher on basic notions from both mereology and ordered sets theory with particular emphasis on supremum relation. We point to some basic properties of supremum which later in the paper we mirror by axioms put on the mereological sum relation and show their consequences. Then we prove that by imposing different requirements on mereological sum we indeed can obtain the equality between sum and supremum. In the final part of the paper we change the perspective and introduce a class of some particular ordered sets that we call mereological posets, in which by a suitable axiom we directly require that sum coincide with supremum. Although we do not give full characterization of such structures we reveal some interesting properties of theirs.