Finite Distributive Lattice Research Articles

Covering Dimension of Finite Distributive Lattices

Covering Dimension of Finite Distributive Lattices

Chain algebras of finite distributive lattices

We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes, we conclude that every such algebra is normal and Cohen–Macaulay, and give an interpretation of its Krull dimension in terms of the combinatorics of the underlying lattice. When the lattice is planar, we show that the corresponding chain algebra is generated by a sortable set of monomials and is isomorphic to a Hibi ring of another finite distributive lattice. As a consequence, it has a defining toric ideal with a quadratic Gröbner basis, and its h-vector counts ascents in certain standard Young tableaux. If instead the lattice has dimension n>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n>2$$\\end{document}, we show that the defining ideal has minimal generators of degree at least n.

Generalized spectra and applications to finite distributive lattices

In our previous paper on frames of continuous functions, the classical adjunction between topological spaces and frames was generalized to a setup in which an arbitrary topological frame replaces the two element chain. The relevant composition of adjoints yields an endofunctor on topological spaces which in general fails to be idempotent. In this paper we prove a formula for iterations of this functor under certain conditions. We apply our result to the construction of finite free distributive lattices and Boolean algebras.

Finite distributive lattices, polyominoes and ideals of König type

Finite distributive lattices whose join-meet ideals are of König type will be classified. Furthermore, a class of polyominoes whose polyomino ideals are of König type will be studied.

A pre-period of a finite distributive lattice

A pre-period of a finite distributive lattice

Absolute Retracts for Finite Distributive Lattices and Slim Semimodular Lattices

Let n denote a positive integer. We describe the absolute retracts for the following five categories of finite lattices: (1) slim semimodular lattices, which were introduced by G. Grätzer and E. Knapp in (Acta. Sci. Math. (Szeged), 73 445–462 2007), and they have been intensively studied since then, (2) finite distributive lattices (3) at most n-dimensional finite distributive lattices, (4) at most n-dimensional finite distributive lattices with cover-preserving {0,1}-homomorphisms, and (5) finite distributive lattices with cover-preserving {0,1}-homomorphisms. Although the singleton lattice is the only absolute retract for the first category, this result has paved the way to some other classes. For the second category, we prove that the absolute retracts are exactly the finite boolean lattices; this generalizes a 1979 result of J. Schmid. For the third category and also for the fourth, the absolute retracts are the finite boolean lattices of dimension at most n and the direct products of n nontrivial finite chains. For the fifth category, the absolute retracts are the same as those for the second category. Also, we point out that in each of these classes, the algebraically closed lattices and the strongly algebraically closed lattices (investigated by J. Schmid and, in several papers, by A. Molkhasi) are the same as the absolute retracts.

The Central Decomposition of FD01(n)

The paper presents a method of composing finite distributive lattices from smaller pieces and applies this to construct the finitely generated free distributive lattices from appropriate Boolean parts.

The fundamental theorem of finite semidistributive lattices

We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff’s Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form “A poset L is a finite semidistributive lattice if and only if there exists a set with some additional structure, such that L is isomorphic to the admissible subsets of ordered by inclusion; in this case, and its additional structure are uniquely determined by L.” The additional structure on is a combinatorial abstraction of the notion of torsion pairs from representation theory and has geometric meaning in the case of posets of regions of hyperplane arrangements. We show how the FTFSDL clarifies many constructions in lattice theory, such as canonical join representations and passing to quotients, and how the semidistributive property interacts with other major classes of lattices. Many of our results also apply to infinite lattices.

Finite distributive nearlattices

Our main goal is to develop a representation for finite distributive nearlattices through certain ordered structures. This representation generalizes the well-known representation given by Birkhoff for finite distributive lattices through finite posets. We also study finite distributive nearlattices through the concepts of dual atoms, boolean elements, complemented elements and irreducible elements. We prove that the sets of boolean elements and complemented elements form semi-boolean algebras. We show that the set of boolean elements of a finite distributive lattice is a boolean lattice.

Fragments of the theory of the enumeration degrees

We prove that every finite distributive lattice can be strongly embedded into the enumeration degrees as an interval, i.e., that there is an interval [a0,a1] of enumeration degrees isomorphic to the lattice, and any enumeration degree b≤a1 lies in this interval or below a0. As corollaries, we conclude that the ∃∀∃-theory of De is undecidable, while the extension of embeddings problem (a subproblem of the ∀∃-theory) is decidable.

Revisiting the representation theorem of finite distributive lattices with principal congruences. A \emph{Proof-by-Picture} approach

Revisiting the representation theorem of finite distributive lattices with principal congruences. A \emph{Proof-by-Picture} approach

Every finite distributive lattice is isomorphic to the minimizer set of an [formula omitted]-concave set function

M♮-concavity is a key concept in discrete convex analysis. For set functions, the class of M♮-concavity is a proper subclass of submodularity. It is a well-known fact that the set of minimizers of a submodular function forms a distributive lattice, where every finite distributive lattice is possible to appear. It is a natural question whether every finite distributive lattice appears as the minimizer set of an M♮-concave set function. This paper affirmatively answers the question.

Cube-free resonance graphs

Let G be a plane elementary bipartite graph with more than two vertices. Then its resonance graph Z(G) is a median graph and the set M(G) of all perfect matchings of G with a specific partial order is a finite distributive lattice. In this paper, we prove that Z(G) is cube-free if and only if it can be obtained from an edge by a sequence of convex path expansions with respect to a reducible face decomposition of G. As a corollary, a structure characterization is provided for G whose Z(G) is cube-free. Furthermore, Z(G) is cube-free if and only if the Clar number of G is at most two, and sharp lower bounds on the number of perfect matchings of G can be expressed by the number of finite faces of G and the number of Clar formulas of G. It is known that a cube-free median graph is not necessarily planar. Using the lattice structure on M(G), we show that Z(G) is cube-free if and only if Z(G) is planar if and only if M(G) is an irreducible sublattice of m×n. We raise a question on how to characterize irreducible sublattices of m×n that are M(G).

On The Representations of Finite Distributive Lattices

On The Representations of Finite Distributive Lattices

Signed ring families and signed posets

ABSTRACT The one-to-one correspondence between finite distributive lattices and finite partially ordered sets (posets) is a well-known theorem of G. Birkhoff. This implies a nice representation of any distributive lattice by its corresponding poset, where the size of the former (distributive lattice) is often exponential in the size of the underlying set of the latter (poset). A lot of engineering and economic applications bring us distributive lattices as a ring family of sets which is closed with respect to the set union and intersection. When it comes to a ring family of sets, the underlying set is partitioned into subsets (or components) and we have a poset structure on the partition. This is a set-theoretical variant of the Birkhoff theorem revealing the correspondence between finite ring families and finite posets on partitions of the underlying sets, which was pursued by Masao Iri around 1978, especially concerned with what is called the principal partition of discrete systems such as graphs, matroids, and polymatroids. In the present paper we investigate a signed-set version of the Birkhoff-Iri decomposition in terms of signed ring family, which corresponds to Reiner's result on signed posets, a signed counterpart of the Birkhoff theorem. We show that given a signed ring family, we have a signed partition of the underlying set together with a signed poset on the signed partition which represents the given signed ring family. This representation is unique up to certain reflections.

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.

English

Distributive lattices form an important, well-behaved class of lattices. They are instances of two larger classes of lattices: congruence-uniform and semidistributive lattices. Congruence-uniform lattices allow for a remarkable second order of their elements: the core label order; semidistributive lattices naturally possess an associated flag simplicial complex: the canonical join complex. In this article we present a characterization of finite distributive lattices in terms of the core label order and the canonical join complex, and we show that the core label order of a finite distributive lattice is always a meet-semilattice.

Compatible Algebras with Straightening Laws on Distributive Lattices

We characterize the finite distributive lattices on which there exists a unique compatible algebra with straightening laws.

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

Let the finite distributive lattice $D$ be isomorphic to the congruence lattice of a finite lattice $L$. Let $Q$ denote those elements of $D$ that correspond to principal congruences under this isomorphism. Then $Q$ contains $0,1 \in D$ and all the join-irreducible elements of $D$. If $Q$ contains exactly these elements, we say that $L$ is a minimal representations of $D$ by principal congruences of the lattice $L$. We characterize finite distributive lattices $D$ with a minimal representation by principal congruences with the property that $D$ has at most two dual atoms.

The cone of supermodular games on finite distributive lattices

In this article we study supermodular functions on finite distributive lattices. Relaxing the assumption that the domain is a powerset of a finite set, we focus on geometrical properties of the polyhedral cone of such functions. Specifically, we generalize the criterion for extremality and study the face lattice of the supermodular cone. An explicit description of facets by the corresponding tight linear inequalities is provided.