scholarly journals Applying the Cz\'edli-Schmidt sequences to congruence properties of planar semimodular lattices

2021 ◽  
Vol 41 (1) ◽  
pp. 153
Author(s):  
G. Gr¤tzer
Keyword(s):  
Semimodular Lattices ◽  
Congruence Properties

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

scholarly journals Algebras Describing Pseudocomplemented, Relatively Pseudocomplemented and Sectionally Pseudocomplemented Posets

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 753
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Universal Algebra ◽  
Relational Structures ◽  
Pseudocomplemented Poset ◽  
The Given ◽  
The Relationship ◽  
Congruence Properties

In order to be able to use methods of universal algebra for investigating posets, we assigned to every pseudocomplemented poset, to every relatively pseudocomplemented poset and to every sectionally pseudocomplemented poset, a certain algebra (based on a commutative directoid or on a λ-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. A certain kind of symmetry can be seen in the relationship between the classes of mentioned posets and the classes of directoids and λ-lattices representing these relational structures. As we show in the paper, this relationship is fully symmetric. Our results show that the assigned algebras satisfy strong congruence properties which can be transferred back to the posets. We also mention applications of such posets in certain non-classical logics.

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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