On the number of join irreducibles and acyclicity in finite modular lattices
Motivated by the relations between modular lattices and geometry, the author introduces the notion of acyclicity for subsets of a finite modular lattice \(L\). It turns out that \(L\) is acyclic if it admits some acyclic base of lines. Let \(d(L)\) and \(s(L)\) be the height of \(L\) and the number of maximal congruences of \(L\), respectively. The main result of the paper is as follows. Some (equivalently: any) base of \(L\) is acyclic if and only if \(L\) has exactly \(2d(L)-s(L)\) join irreducibles.
scientific article; zbMATH DE number 5886745
scientific article