Semimodular Lattices

2016 ◽  
pp. 165-172
Author(s):  
George Grätzer
Keyword(s):  
Semimodular Lattices

scholarly journals A note on fixed points in semimodular lattices

1980 ◽  
Vol 29 (3) ◽  
pp. 245-250 ◽  
Author(s):  
Anders Björner ◽  
Ivan Rival
Keyword(s):  
Fixed Points ◽  
Semimodular Lattices

Matchings and Radon transforms in lattices II. Concordant sets

1987 ◽  
Vol 101 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Incidence Matrix ◽  
Finite Lattice ◽  
Möbius Function ◽  
Radon Transforms ◽  
Modular Lattices ◽  
Covering Theorem ◽  
Mobius Function ◽  
Finite Lattices ◽  
Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

scholarly journals Congruence Lattices of Finite Semimodular Lattices

1998 ◽  
Vol 41 (3) ◽  
pp. 290-297 ◽  
Author(s):  
G. Grätzer ◽  
H. Lakser ◽  
E. T. Schmidt
Keyword(s):  
Distributive Lattice ◽  
Congruence Lattice ◽  
Finite Distributive Lattice ◽  
Congruence Lattices ◽  
Semimodular Lattices

AbstractWe prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) semimodular lattice.

scholarly journals Representations of Matroids in Semimodular Lattices

2001 ◽  
Vol 22 (6) ◽  
pp. 789-799 ◽  
Author(s):  
Alexandre V. Borovik ◽  
Israel M. Gelfand ◽  
Neil White
Keyword(s):  
Semimodular Lattices

scholarly journals Congruence structure of planar semimodular lattices: the General Swing Lemma

2018 ◽  
Vol 79 (2) ◽  
Author(s):  
Gábor Czédli ◽  
George Grätzer ◽  
Harry Lakser
Keyword(s):  
Semimodular Lattices ◽  
Congruence Structure
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