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

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...

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Bibliographic Details
Published in:Order (Dordrecht) 2023-07, Vol.40 (2), p.403-421
Main Author: Czédli, Gábor
Format: Article
Language:English
Subjects:
Algebra
Discrete Mathematics
Lattice modules
Lattices
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
Theorems
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Summary: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 Math. 225 , 2455–2463, 2010 ) on extensions to geometric lattices, and an unpublished result of E. T. Schmidt.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.
ISSN:0167-8094
1572-9273
DOI:10.1007/s11083-022-09620-8