  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.
