A new look at the Jordan-Hölder theorem for semimodular lattices
The authors prove the following theorem. Let \(C\) and \(D\) be chains in a semimodular lattice \(L\), where \(C: 0=c_0\prec c_1\prec\dots\prec c_n=1\) and \(D:0=d_0\prec d_1\prec\dots\prec d_n=1\). Then there is a permutation \(\pi\) of the set \(\{1,2,\dots,n\}\) with the following property: for each \(i\), there exists a prime interval \(p_i\) in \(L\) such that \([c_{i-1}, c_i]\) is up-perspective to \(p_i\) and \(p_i\) is down-perspective to \([d_{\pi(i)-1}, d_{\pi(i)}]\). This sharpens a well-known result on semimodular lattices [cf., e.g., \textit{G. Birkhoff}, Lattice theory. Third (new) ed. Providence, R.I.: American Mathematical Society (1967; Zbl 0153.02501)].
scientific article; zbMATH DE number 5886737