Orthomodular Lattices Induced by the Concurrency Relation

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