Congrunce lattices of rectangular lattices1

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.

Lattice Bases of Planar Lattices

In this paper we give a characterization of planar lattices, and show that the sublattice [Formula: see text] corresponding to the Hibi ideal of a finite distributive lattice [Formula: see text] has a nice lattice basis if and only if [Formula: see text] is planar. We also consider the Gröbner basis of the lattice basis ideal attached to the planar lattices.

Lattices Related to Extensions of Presentations of Transversal Matroids

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 $\{1,2,\ldots,r(M)\}$.

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

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.

Reimer's Inequality on a Finite Distributive Lattice

We generalize Reimer's Inequality [6] (a.k.a. the BKR Inequality or the van den Berg–Kesten Conjecture [1]) to the setting of finite distributive lattices.

Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order

Let L be a finite distributive lattice. Let Sub0(L) be the lattice﹛S | S is a sublattice of L﹜ [ ﹛∅﹜ and let ℓ*[Sub0(L)] be the length of the shortest maximal chain in Sub0(L). It is proved that if K and L are non-trivial finite distributive lattices, then ℓ*[Sub0(K × L)] = ℓ*[Sub0(K)] + ℓ[Sub0(L)]. A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved.