Jordan-H\"older with uniqueness for semimodular semilattices
Combinatorics
2019-09-20 v2
Abstract
We present a short proof of the Jordan-H\"older 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.
Keywords
Cite
@article{arxiv.1908.09912,
title = {Jordan-H\"older with uniqueness for semimodular semilattices},
author = {Pavel Paták},
journal= {arXiv preprint arXiv:1908.09912},
year = {2019}
}