Length-preserving Extensions of a Semimodular Lattice by Lowering a Join-irreducible Element

Abstract

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. As the main application, 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. This theorem implies earlier results proved by Grätzer and Kiss (Order 2 351–365, 1986), Wild (Discrete Math. 112, 207–244, 1993), and Czédli and Schmidt (Adv. Math. 225, 2455–2463, 2010) on extensions to geometric lattices, and an unpublished result of E. T. Schmidt. Our approach offers shorter proofs of these results than the original ones.