Closure operators associated to partially ordered sets

Partially ordered sets are a natural framework in which the semantics of concurrent processes can be defined and studied. From a partial order, two interesting binary relations can be immediately defined: one can be interpreted as a causal dependence, the other as causal independence. Both are symmetric, but in general non transitive. By applying standard techniques in lattice theory, one can then derive from each of them a closure operator on the underlying set of the partial order, and a corresponding complete lattice, whose elements are the closed subsets of the partially ordered set. In a recent paper, we applied this idea to the independence (or concurrency) relation; in this paper we deal with the dependence relation; some structural properties of the corresponding closed sets are given, and a subclass of closed sets, called spatially closed sets, is identified. The main result states that this subclass forms an algebraic lattice.