Semimodularity and the Jordan–Hölder theorem in posets, with applications to partial partitions

Abstract

Lattice-theoretical generalizations of the Jordan–Holder theorem of group theory give isomorphisms between finite maximal chains with same endpoints. The best one has been given by Czedli and Schmidt (after Gratzer and Nation), and 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 (Bull Amer Math Soc 49(8):567–568, 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–Holder 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.