The matrix of a slim semimodular lattice
Slim semimodular lattices are particular planar lattices: they do not admit any 3-element antichain of join irreducibles. The first main result of the paper under review establishes a 1-1-correspondence between such lattices, which are not decomposable as the gluing of two intervals, and certain join congruences on the direct product of two chains. The latter, again, are shown to be in 1-1-correspondence with certain subsets of intervals, isomorphic to \(\mathbf{2}\times \mathbf{2}\), of these direct products -- also to be described in terms of \(0\)-\(1\)-matrices. Background of these investigations is a detailed study of the lattice theory behind the Jordan-Hölder theorem, given by the author and \textit{E. T. Schmidt} in [Algebra Univers. 66, No. 1--2, 69--79 (2011; Zbl 1233.06006)].
scientific article; zbMATH DE number 6018078