Revisiting Faigle geometries from a perspective of semimodular lattices

TL;DR

This paper revisits Faigle geometries, simplifies their axiomatization, and demonstrates their usefulness by providing concise proofs of key theorems on extensions of finite semimodular lattices.

Contribution

It offers a simplified axiomatization of Faigle geometries and applies them to prove extension theorems for finite semimodular lattices.

Findings

Simplified axiomatization of Faigle geometries.

Short proof that slim semimodular lattices extend to slim rectangular lattices.

Short proof that finite semimodular lattices extend to geometric lattices.

Abstract

In 1980, U. Faigle introduced a sort of finite geometries on posets that are in bijective correspondence with finite semimodular lattices. His result has almost been forgotten in lattice theory. Here we simplify the axiomatization of these geometries, which we call Faigle geometries. To exemplify their usefulness, we give a short proof of a theorem of Gr\"atzer and E. Knapp (2009) asserting that each slim semimodular lattice $L$ has a congruence-preserving extension to a slim rectangular lattice of the same length as $L$. As another application of Faigle geometries, we give a short proof of G. Gr\"atzer and E. W. Kiss' result from 1986 (also proved by M. Wild in 1993 and the present author and E. T. Schmidt in 2010) that each finite semimodular lattice $L$ has an extension to a geometric lattice of the same length as $L$.