An Analogue of Covering Space Theory for Ranked Posets

  • Michael E. Hoffman


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 $\tilde P\to P$, unique up to isomorphism, so that
1. $\tilde P\to P$ factors through any other covering map $P'\to P$;
2. every principal order ideal of $\tilde P$ is a chain; and
3. the weight assigned to each covering relation of $\tilde P$ is 1. If $P$ is a poset of "natural" combinatorial objects, the elements of its universal cover $\tilde P$ 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 $\tilde P$ is the poset of standard Young tableaux; if $P$ is the poset of rooted trees ordered by inclusion, then $\tilde P$ consists of permutations. We discuss several other examples, including the posets of necklaces, bracket arrangements, and compositions.

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