tailieunhanh - Báo cáo toán học: "An Analogue of Covering Space Theory for Ranked Posets"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: An Analogue of Covering Space Theory for Ranked Posets. | An Analogue of Covering Space Theory for Ranked Posets Michael E. Hoffman Dept. of Mathematics U. S. Naval Academy Annapolis MD 21402 meh@ Submitted May 10 2001. Accepted October 11 2001. MR Classifications Primary 06A07 05A15 Secondary 57M10 Abstract 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 P P unique up to isomorphism so that 1. P P factors through any other covering map P P 2. every principal order ideal of P is a chain and 3. the weight assigned to each covering relation of P is 1. If P is a poset of natural combinatorial objects the elements of its universal cover 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 P is the poset of standard Young tableaux if P is the poset of rooted trees ordered by inclusion then P consists of permutations. We discuss several other examples including the posets of necklaces bracket arrangements and compositions. 1 Introduction For topological spaces the notion of a covering space is familiar see . 9 a covering map p X X is a continuous surjection such that for sufficiently small open sets U c X p-1 U is a disjoint union of open sets in X0 each of which p maps homeomorphically onto U. For any space X satisfying appropriate hypotheses . that X is connected locally arcwise connected and semilocally simply connected there is a simply connected covering space X X which is universal in the sense that it factors through any other connected cover of X . if p X0 X is any covering map with X0 conneced THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R32 1 then there is a covering map f X X so that .