Length-preserving extensions of semimodular lattices by lowering join-irreducible elements

TL;DR

This paper introduces a method to extend semimodular lattices while preserving length, allowing for the lowering of join-irreducible elements, and provides shorter proofs for several known extension results.

Contribution

It develops a length-preserving extension technique for semimodular lattices, generalizes the correspondence with Faigle geometries, and simplifies proofs of existing theorems.

Findings

Established a lowering construction for join-irreducible elements.

Proved a general theorem on length-preserving extensions.

Unified and shortened proofs of previous extension results.

Abstract

We prove that if $e$ is a join-irreducible element of a semimodular lattice $L$ of finite length and $h<e$ in $L$ such that $e$ does not cover $h$, then $e$ can be "lowered" to a covering of $h$ by taking a length-preserving semimodular extension $K$ of $L$ but not changing the rest of join-irreducible elements. With the help of our "lowering construction", we prove a general theorem on length-preserving semimodular extensions of semimodular lattices, which implies some earlier results proved by G. Gr\"atzer and E. W. Kiss (1986), M. Wild (1993), and G. Cz\'edli and E. T. Schmidt (2010) on extensions to geometric lattices, and even an unpublished result of E. T. Schmidt on higher dimensional rectangular lattices. Our method offers shorter proofs of these results than the original ones. To obtain the main tool used in the paper, we extend the bijective correspondence between finite semimodular lattices and Faigle geometries to an analogous correspondence between semimodular lattices of finite lengths and a larger class of geometries.