Orthomodular Lattices Induced by the Concurrency Relation

We apply to locally finite partially ordered sets a construction which associates a complete lattice to a given poset; the elements of the lattice are the closed subsets of a closure operator, defined starting from the concurrency relation. We show that, if the partially ordered set satisfies a property of local density, i.e.: N-density, then the associated lattice is also orthomodular. We then consider occurrence nets, introduced by C.A. Petri as models of concurrent computations, and define a family of subsets of the elements of an occurrence net; we call those subsets "causally closed" because they can be seen as subprocesses of the whole net which are, intuitively, closed with respect to the forward and backward local state changes. We show that, when the net is K-dense, the causally closed sets coincide with the closed sets induced by the closure operator defined starting from the concurrency relation. K-density is a property of partially ordered sets introduced by Petri, on the basis of former axiomatizations of special relativity theory.