On the purity of the lattice of sublattices of a finite distributive lattice

1982 ◽  
Vol 15 (1) ◽  
pp. 258-271
Author(s):  
C. C. Chen ◽  
K. M. Koh ◽  
S. C. Lee
Keyword(s):  
Distributive Lattice ◽  
Finite Distributive Lattice

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 The Lattice of Definable Equivalence Relations in Homogeneous $n$-Dimensional Permutation Structures

10.37236/5980 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Samuel Braunfeld
Keyword(s):  
Distributive Lattice ◽  
Equivalence Relations ◽  
Electronic Journal ◽  
Finite Distributive Lattice ◽  
Linear Orders ◽  
Finite Dimensional ◽  
Homogeneous Structures ◽  
Special Case

In Homogeneous permutations, Peter Cameron [Electronic Journal of Combinatorics 2002] classified the homogeneous permutations (homogeneous structures with 2 linear orders), and posed the problem of classifying the homogeneous $n$-dimensional permutation structures (homogeneous structures with $n$ linear orders) for all finite $n$. We prove here that the lattice of $\emptyset$-definable equivalence relations in such a structure can be any finite distributive lattice, providing many new imprimitive examples of homogeneous finite dimensional permutation structures. We conjecture that the distributivity of the lattice of $\emptyset$-definable equivalence relations is necessary, and prove this under the assumption that the reduct of the structure to the language of $\emptyset$-definable equivalence relations is homogeneous. Finally, we conjecture a classification of the primitive examples, and confirm this in the special case where all minimal forbidden structures have order 2. 

scholarly journals Every finite distributive lattice is a set of stable matchings for a small stable marriage instance

1987 ◽  
Vol 44 (2) ◽  
pp. 304-309 ◽  
Author(s):  
Dan Gusfield ◽  
Robert Irving ◽  
Paul Leather ◽  
Michael Saks
Keyword(s):  
Distributive Lattice ◽  
Stable Marriage ◽  
Finite Distributive Lattice ◽  
Stable Matchings

scholarly journals Every finite distributive lattice is a set of stable matchings

1984 ◽  
Vol 37 (3) ◽  
pp. 353-356 ◽  
Author(s):  
Charles Blair
Keyword(s):  
Distributive Lattice ◽  
Finite Distributive Lattice ◽  
Stable Matchings

Congrunce lattices of rectangular lattices1

2020 ◽  
Vol 39 (3) ◽  
pp. 2831-2843
Author(s):  
Peng He ◽  
Xue-Ping Wang
Keyword(s):  
Distributive Lattice ◽  
Upper Bound ◽  
Congruence Lattice ◽  
Semimodular Lattice ◽  
Rectangular Lattice ◽  
Finite Distributive Lattice ◽  
Irreducible Elements

Let D be a finite distributive lattice with n join-irreducible elements. It is well-known that D can be represented as the congruence lattice of a rectangular lattice L which is a special planer semimodular lattice. In this paper, we shall give a better upper bound for the size of L by a function of n, improving a 2009 result of G. Grätzer and E. Knapp.

scholarly journals On the Strong Purity of the Sublattice-Lattice of a Finite Distributive Lattice

1983 ◽  
Vol 06 (2) ◽  
pp. 381-388
Author(s):  
C. C. CHEN ◽  
K. M. KOH
Keyword(s):  
Distributive Lattice ◽  
Finite Distributive Lattice

scholarly journals Lattices Related to Extensions of Presentations of Transversal Matroids

10.37236/5466 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Joseph E. Bonin
Keyword(s):  
Distributive Lattice ◽  
Upper Bound ◽  
Closure Operator ◽  
Ordered Set ◽  
Main Tool ◽  
Upper Bounds ◽  
Single Element ◽  
Finite Distributive Lattice ◽  
Closed Sets ◽  
Transversal Matroid

For a presentation $\mathcal{A}$ of a transversal matroid $M$, we study the ordered set $T_{\mathcal{A}}$ of single-element transversal extensions of $M$ that have presentations that extend $\mathcal{A}$; extensions are ordered by the weak order.  We show that $T_{\mathcal{A}}$ is a distributive lattice, and that each finite distributive lattice is isomorphic to $T_{\mathcal{A}}$ for some presentation $\mathcal{A}$ of some transversal matroid $M$. We show that $T_{\mathcal{A}}\cap T_{\mathcal{B}}$, for any two presentations $\mathcal{A}$ and $\mathcal{B}$ of $M$, is a sublattice of both $T_{\mathcal{A}}$ and $T_{\mathcal{B}}$. We prove sharp upper bounds on $|T_{\mathcal{A}}|$ for presentations $\mathcal{A}$ of rank less than $r(M)$ in the order on presentations; we also give a sharp upper bound on $|T_{\mathcal{A}}\cap T_{\mathcal{B}}|$. The main tool we introduce to study $T_{\mathcal{A}}$ is the lattice $L_{\mathcal{A}}$ of closed sets of a certain closure operator on the lattice of subsets of $\{1,2,\ldots,r(M)\}$.

scholarly journals Minimal representations of a finite distributive lattice by principal congruences of a lattice

2019 ◽  
Vol 85 (12) ◽  
pp. 69-96
Author(s):  
George Grätzer ◽  
Harry Lakser
Keyword(s):  
Distributive Lattice ◽  
Finite Distributive Lattice ◽  
Minimal Representations

Homomorphisms of Distributive Lattices as Restrictions of Congruences

1986 ◽  
Vol 38 (5) ◽  
pp. 1122-1134 ◽  
Author(s):  
George Grätzer ◽  
Harry Lakser
Keyword(s):  
Distributive Lattice ◽  
Congruence Lattice ◽  
Classical Result ◽  
Finite Lattice ◽  
Distributive Lattices ◽  
Lattice Homomorphism ◽  
Finite Distributive Lattice ◽  
Full Generality ◽  
Ideal Lattices

Given a lattice L and a convex sublattice K of L, it is well-known that the map Con L → Con K from the congruence lattice of L to that of K determined by restriction is a lattice homomorphism preserving 0 and 1. It is a classical result (first discovered by R. P. Dilworth, unpublished, then by G. Grätzer and E. T. Schmidt [2], see also [1], Theorem II.3.17, p. 81) that any finite distributive lattice is isomorphic to the congruence lattice of some finite lattice. Although it has been conjectured that any algebraic distributive lattice is the congruence lattice of some lattice, this has not yet been proved in its full generality. The best result is in [4]. The conjecture is true for ideal lattices of lattices with 0; see also [3].

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