Cutset condition for geometric lattices
Let \(L\) be a lattice with bottom \(0\) and top \(1\), \(a, b \in L\), \(aMb\) will denote that \((a,b)\) is a modular pair. If \(aMx\) and \(xMa\) holds for any \(x\in L\), then \(a\) is called a modular element. If a subset \(C\) of \(L\) meets every maximal chain, then \(C\) is called a cutset of \(L\). The modular cutset condition requires that there exists a cutset \(C\) of modular elements such that \(0\not\in C, 1 \not\in C\). The authors continue their work on cutsets and geometric lattices. Their main result is: If an atomistic lattice \(L\) of finite length fulfils the modular cutset condition, then \(L\) is semimodular.
geometric lattice, semimodular lattice of finite length
scientific article; zbMATH DE number 5146177