An Analogue of Covering Space Theory for Ranked Posets

Suppose P is a partially ordered set that is locally finite, has a least element, and admits a rank function. We call P a weighted-relation poset if all the covering relations of P are assigned a positive integer weight. We develop a theory of covering maps for weighted-relation posets, and in particular show that any weighted-relation poset P has a universal cover P̃ → P , unique up to isomorphism, so that 1. P̃ → P factors through any other covering map P ′ → P ; 2. every principal order ideal of P̃ is a chain; and 3. the weight assigned to each covering relation of P̃ is 1. If P is a poset of “natural” combinatorial objects, the elements of its universal cover P̃ often have a simple description as well. For example, if P is the poset of partitions ordered by inclusion of their Young diagrams, then the universal cover P̃ is the poset of standard Young tableaux; if P is the poset of rooted trees ordered by inclusion, then P̃ consists of permutations. We discuss several other examples, including the posets of necklaces, bracket arrangements, and compositions.