2019/08/26 by Pavel Paták, Paták, Pavel
Mathematics · #05B35 #06C10 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05B35 #msc:06C10
paper · pdf · doi:10.48550/arxiv.1908.09912
arxiv created 2019/09/19 · arxiv updated 2019/09/20
We present a short proof of the Jordan-Hölder theorem with uniqueness for semimodular semilattice: Given two maximal chains in a semimodular semilattice of finite height, they both have the same length. Moreover there is a unique bijection that takes the prime intervals of the first chain to the prime intervals of the second chain such that the interval and its image are up-and-down projective. The theorem generalizes the classical result that all composition series of a finite group have the same length and isomorphic factors. Moreover, it shows that the isomorphism is in some sense unique.