Article in volume
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Title:
Revisiting Faigle geometries from a perspective of semimodular lattices
PDFSource:
Discussiones Mathematicae - General Algebra and Applications 43(2) (2023) 207-222
Received: 2021-07-22 , Revised: 2021-11-30 , Accepted: 2021-11-30 , Available online: 2023-01-11 , https://doi.org/10.7151/dmgaa.1416
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
Grätzer 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. Grätzer and E.W. Kiss' result from 1986 (also proved by
M. Wild in 1993, the present author and E.T. Schmidt in 2010, and B. Skublics
in 2013) that each finite semimodular lattice $L$ has an extension to
a geometric lattice of the same length as $L$.
Keywords:
Faigle geometry, semimodular lattice, planar semimodular lattice, rectangular lattice, congruence-preserving extension, slim semimodular lattice, geometric lattice, cover-preserving extension
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