Orthomodular algebraic lattices related to combinatorial posets

We extend some theoretical results in the frame of concurrency theory, which were presented in [1]. In particular, we focus on partially ordered sets (posets) as models of nonsequential processes [2] and we apply the same construction as in [1] of a lattice of subsets of points of the poset via a closure operator defined on the basis of the concurrency relation, viewed as lack of causal dependence. The inspiring idea is related to works by C. A. Petri [6]. Petri proposed a theory of systems based on abstract models to represent the behaviour and the properties of concurrent and distributed systems, which takes into account the principles of the special relativity. A crucial difference between the standard physical theories and the framework in which Petri develops his own theory comes from the use of the continuum as the underlying model in physics. In the combinatorial model proposed by Petri, the usual notion of density of the continuum model is replaced by two properties strictly related and required for the posets modelling a discrete space-time: the so-called K-density and a weaker form called N-density. K-density is based on the idea that any maximal antichain (or cut) in a poset and any maximal chain (or line) have a nonempty intersection. A line can be interpreted as a sequential subprocess, while a cut corresponds to a time instant and K-density requires that, at any time instant, any sequential subprocess must be in some state or changing its state. N-density can be viewed as a sort of local density and was introduced by Petri as an axiom for posets modelling nonsequential processes. Occurrence nets, a fundamental model of such processes, are indeed N-dense, whereas for example event structures [5] are in general not N-dense. In [1] we have considered as model of non sequential processes a class of locally finite posets and shown that the closed subsets, obtained via a closure operator defined on the basis of concurrency, correspond in general to subprocesses which result to be ‘closed’ with respect to the Petri net firing rule. Moreover, we have shown that if the poset is N-dense, then the lattice of closed subsets is orthomodular. Orthomodular lattices are families of partially overlapping Boolean algebras and have been studied as the algebraic model of quantum logic [7].