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.

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.

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 It is a classical result (first discovered by R. P. Dilworth, unpublished, then by G. Grätzer and E. T. Schmidt [2], see also [1], Theorem II.3.17, p. 81) that any finite distributive lattice is isomorphic to the congruence lattice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].

scholarly journals On the independence theorem of related structures for modular (arguesian) lattices

2003 ◽  
Vol 40 (1-2) ◽  
pp. 1-12 ◽  
Author(s):  
G. Grätzer ◽  
E. T. Schmidt
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Distributive Lattice ◽  
Congruence Lattice ◽  
Finite Distributive Lattice ◽  
Arguesian Lattice ◽  
A Finite Group ◽  
Independence Theorem

Let D be a finite distributive lattice with more than one element and let G be a finite group. We prove that there exists a modular (arguesian) lattice M such that the congruence lattice of M is isomorphic to D and the automorphism group of M is isomorphic to G.

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 A special sublattice of the congruence lattice of a regular semigroup

1997 ◽  
Vol 40 (3) ◽  
pp. 457-472 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Distributive Lattice ◽  
Regular Semigroup ◽  
Congruence Lattice ◽  
Regular Semigroups ◽  
Generators And Relations ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Special Cases ◽  
Completely Regular Semigroups ◽  
Completely Simple

Let S be a regular semigroup and be its congruence lattice. For ρ ∈ , we consider the sublattice Lρ of generated by the congruences pw where w ∈ {K, k, T, t}* and w has no subword of the form KT, TK, kt, tk. Here K, k, T, t are the operators on induced by the kernel and the trace relations on . We find explicitly the least lattice L whose homomorphic image is Lρ for all ρ ∈ and represent it as a distributive lattice in terms of generators and relations. We also consider special cases: bands of groups, E-unitary regular semigroups, completely simple semigroups, rectangular groups as well as varieties of completely regular semigroups.

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