Determination of a poset by its co-relation

Abstract

The co-relation of a partially ordered set (poset) holds true between a and b iff neither a<b nor b<a. Since the co-relation does not change if we proceed to the dual poset, the co-relation determines the order relation at most up to duality. This remark leads us to the idea of "natural order": A poset is called a natural order iff the order relation is determined uniquely up to duality by the co-relation. If P is a poset consisting of three elements a, b, and c with a<b, a<c, and neither b<c nor c<b, then (P,≤) is a natural order, the underlying undirected graph of which forms a v-shaped figure. A poset is called "v-connected" iff any two edges in the underlying undirected graph can be "connected" by a sequence of v-shaped figures. It will be proved that a poset is a natural order iff it is v-connected. Applying this result to occurrence posets (a special kind of posets corresponding to occurrence nets) it will turn out that connected occurrence posets with the property that all of its elements have only a finite number of predecessors and successors are natural orders iff some simple, "locally" verifiable conditions hold true.