Orthomodular Lattices in Occurrence Nets

Abstract

In this paper, we study partially ordered structures associated to occurrence nets. An occurrence net is endowed with a symmetric, but in general non transitive, concurrency relation. By applying known techniques in lattice theory, from any such relation one can derive a closure operator, and then an orthocomplemented lattice. We prove that, for a general class of occurrence nets, those lattices, formed by closed subsets of net elements, are orthomodular. A similar result was shown starting from a simultaneity relation defined, in the context of special relativity theory, on Minkowski spacetime. We characterize the closed sets, and study several properties of lattices derived from occurrence nets; in particular we focus on properties related to K-density. We briefly discuss some variants of the construction, showing that, if we discard conditions, and only keep the partial order on events, the corresponding lattice is not, in general, orthomodular.