A NEW LOOK AT THE SEMIMODULAR LATTICES A GEOMETRIC APPROACH

Abstract

Abstract. This paper is an experiment, how it is possible to treat semimodular lattices as geometric shapes. In 2009 I published a paper with Gábor Czédli, [4], we proved that every semimodular lattice L can be obtained from a direct power of a chain G = Cn – geometrically a cube – on an easy way. L is the cover-preserving join-homomorphism of G. We introduce the concept of rectangular semimodular lattices and prove that the building stones of semimodular lattices are special rectangulars, the pigeonholes. The building tool is a special S-verklebte sum (introduced by Christian Herrmann [21]) this is the patchwork. This manuscript includes parts of the papers [2], [3], [4], [5], [6], [7], [8], [9],