Semimodularity and the Jordan–Hölder theorem in posets, with applications to partial partitions
Résumé
Lattice-theoretical generalizations of the Jordan-H\"older theorem of group theory give isomorphisms between finite maximal chains with same endpoints. The best one has been given by Cz{\'e}dli and Schmidt (after Gr\"atzer and Nation), it applies to semimodular lattices and gives a chain isomorphism by iterating up and down the perspectivity relation between intervals $[x\wedge y,x]$ and $[y,x\vee y]$ where $x$ covers $x\wedge y$ and $x\vee y$ covers $y$. In this paper we extend to arbitrary (and possibly infinite) posets the definitions of standard semimodularity and of the slightly weaker ``Birkhoff condition'', following the approach of Ore (1943). Instead of perspectivity, we associate tags to the covering relation, a more flexible approach. We study the finiteness and length constancy of maximal chains under both conditions, and obtain Jordan-H\"older theorems. Our theory is easily applied to groups, to closure ranges of an arbitrary poset, and also to five new order relations on the set of partial partitions of a set (i.e., partitions of its subsets), which do not constitute lattices.
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