Modular lattices of finite length (Part B)

TL;DR

This paper explores geometric representations of finite-length modular lattices using bases of lines, providing bounds on join-irreducible elements and an algorithm for submodule lattice computation.

Contribution

It introduces the concept of bases of lines for modular lattices and presents an algorithm to compute submodule lattices efficiently.

Findings

Finite-length modular lattices have at least 2d(L)-s(L) join-irreducible elements.

The paper proposes an algorithm for calculating submodule lattices of finite modules.

Bases of lines offer a geometric perspective on modular lattices.

Abstract

Part B (of a project involving four Parts) is about "bases of lines", a concept introduced by C. Herrmann and the author in the late 80's. Bases of lines attempt to describe a given modular lattice in a geometric way akin to how projective geometries describe complemented modular lattices. This e.g. yields the result that each modular lattice L of finite length d(L), and having s(L) many maximal congruences, has at least 2d(L)-s(L) many join-irreducible elements. Furthermore, an algorithm is proposed that calculates, in a compressed way, the (full) submodule lattice Sub(W) of any (sufficiently known) finite R-module W.