Semimodular Lattices Research Articles

Modularity conditions in Leibniz algebras

In this paper, we continue the study of the subalgebra lattice of a Leibniz algebra. In particular, we find out that solvable Leibniz algebras with an upper semi-modular lattice are either almost-abelian or have an abelian ideal spanned by the elements with square zero. We also study Leibniz algebras in which every subalgebra is a weak quasi-ideal, as well as modular symmetric Leibniz algebras.

Union-closed sets and Horn Boolean functions

A family F of sets is union-closed if the union of any two sets from F belongs to F. The union-closed sets conjecture states that if F is a finite union-closed family of finite sets, then there is an element that belongs to at least half of the sets in F. The conjecture has several equivalent formulations in terms of other combinatorial structures such as lattices and graphs. In its whole generality the conjecture remains wide open, but it was verified for various important classes of lattices, such as lower semimodular lattices, and graphs, such as chordal bipartite graphs. In the present paper we develop a Boolean approach to the conjecture and verify it for several classes of Boolean functions, such as submodular functions and double Horn functions.

Infinitely many new properties of the congruence lattices of slim semimodular lattices

Slim planar semimodular lattices (SPS lattices or slim semimodular lattices for short) were introduced by G. Grätzer and E. Knapp in 2007. More than four dozen papers have been devoted to these (necessarily finite) lattices and their congruence lattices since then. In addition to distributivity, the congruence lattices of SPS lattices satisfy seven known properties. Out of these seven properties, the first two were published by G. Grätzer in 2016 and 2020, the next four by the present author in 2021, and the seventh jointly by G. Grätzer and the present author in 2022. Here we give two infinite families of new properties of the congruence lattices of SPS lattices. These properties are independent. We also present stronger versions of these properties but not all of them are independent, and improve three out of the seven previously known properties. The approach is based on lamps, which we introduced in a 2021 paper. In addition to using the 2021 results, we need to prove some easy new lemmas on lamps.

The Best Extending Cover-preserving Geometric Lattices of Semimodular Lattices

The Best Extending Cover-preserving Geometric Lattices of Semimodular Lattices

On slim rectangular lattices

Let L be a slim, planar, semimodular lattice (slim means that it does not contain an $${{\textsf{M}}}_3$$ -sublattice). We call the interval $$I = [o, i]$$ of L rectangular, if there are complementary $$a, b \in I$$ such that a is to the left of b. We claim that a rectangular interval of a slim rectangular lattice is also a slim rectangular lattice. We will present some applications, including a recent result of G. Czédli. In a paper with E. Knapp about a dozen years ago, we introduced natural diagrams for slim rectangular lattices. Five years later, G. Czédli introduced $${\mathcal {C}}_1$$ -diagrams. We prove that they are the same.

Revisiting Faigle geometries from a perspective of semimodular lattices

Revisiting Faigle geometries from a perspective of semimodular lattices

Length-preserving Extensions of a Semimodular Lattice by Lowering a Join-irreducible Element

Length-preserving Extensions of a Semimodular Lattice by Lowering a Join-irreducible Element

A property of meets in slim semimodular lattices and its application to retracts

Slim semimodular lattices were introduced by G. Grätzer and E. Knapp in 2007, and they have intensively been studied since then. These lattices can be given by \(\mathcal{C}_1\)-diagrams, defined by the author in 2017. We prove that if x and y are incomparable elements in such a lattice L, then their meet has the property that the interval \([x \wedge y, x]\) is a chain, this chain is of a normal slope in every \(\mathcal{C}_1\)-diagram of L, and except possibly for x, the elements of this chain are meet-reducible.In the direct square K1 of the three-element chain, let X1 and A1 be the set of atoms and the sublattice generated by 0 and the coatoms, respectively. Denote by K2 the unique eight-element lattice embeddable in K1. Let A2 be the sublattice of K2 consisting of 0, 1, the meet-reducible atom, and the join-reducible coatom. Let X2 stand for the singleton consisting of the doubly reducible element of K2. For i = 1, 2, we apply the above-mentioned property of meets to prove that whenever Ki is a sublattice and Si is a retract of a slim semimodular lattice, then \(A_i \subseteq S_i\) implies that \(X_i \subseteq S_i\).

Semi-complements and complements in semi-modular lattices

Semi-complements and complements in semi-modular lattices

Contributions to the theory of semimodular lattices

Contributions to the theory of semimodular lattices

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.

A new property of congruence lattices of slim, planar, semimodular lattices

A new property of congruence lattices of slim, planar, semimodular lattices

Cyclic congruences of slim semimodular lattices and non-finite axiomatizability of some finite structures

We give a new proof of the fact that finite bipartite graphs cannot be axiomatized by finitely many first-order sentences among finite graphs. (This fact is a consequence of a general theorem proved by L. Ham and M. Jackson, and the counterpart of this fact for all bipartite graphs in the class of all graphs is a well-known consequence of the compactness theorem.) Also, to exemplify that our method is applicable in various fields of mathematics, we prove that neither finite simple groups, nor the ordered sets of join-irreducible congruences of slim semimodular lattices can be described by finitely many axioms in the class of finite structures. Since a 2007 result of G. Grätzer and E. Knapp, slim semimodular lattices have constituted the most intensively studied part of lattice theory and they have already led to results even in group theory and geometry. In addition to the non-axiomatizability results mentioned above, we present a new property, called Decomposable Cyclic Elements Property, of the congruence lattices of slim semimodular lattices.

Rank axiom of modular supermatroids: A connection with directional DR submodular functions

A matroid has been one of the most important combinatorial structures since it was introduced by Whitney as an abstraction of linear independence. As an important property of a matroid, it can be characterized by several different (but equivalent) axioms, such as the augmentation, the base exchange, or the rank axiom.A supermatroid is a generalization of a matroid defined on lattices. Here, the central question is whether a supermatroid can be characterized by several equivalent axioms similar to a matroid. Barnabei, Nicoletti, and Pezzoli characterized supermatroids on distributive lattices, and Fujishige, Koshevoy, and Sano generalized the results for cg-matroids (supermatroids on lower locally distributive lattices).In this study, we focus on modular lattices, which are an important superclass of distributive lattices, and provide equivalent characterizations of supermatroids on modular lattices. We characterize supermatroids on modular lattices using the rank axiom in which the rank function is a directional DR-submodular function, which is a generalization of a submodular function introduced by the authors. Using a characterization based on rank functions, we further prove the strong exchange property of a supermatroid, which has application in optimization.We also reveal the relation between the axioms of a supermatroid on lower semimodular lattices, which is a common superclass of a lower locally distributive lattice and a modular lattice.

Lamps in slim rectangular planar semimodular lattices

A planar (upper) semimodular lattice $L$ is slim if the five-element nondistributive modular lattice $M_3$ does not occur among its sublattices. (Planar lattices are finite by definition.) Slim rectangular lattices as particular slim planar semimodular lattices were defined by G. Gr\atzer and E. Knapp in 2007. In 2009, they also proved that the congruence lattices of slim planar semimodular lattices with at least three elements are the same as those of slim rectangular lattices. In order to provide an effective tool for studying these congruence lattices, we introduce the concept of lamps of slim rectangular lattices and prove several of their properties. Lamps and several tools based on them allow us to prove in a new and easy way that the congruence lattices of slim planar semimodular lattices satisfy the two previously known properties. Also, we use lamps to prove that these congruence lattices satisfy four new properties including the two-pendant four-crown property and the forbidden marriage property.

The median partition and submodularity

A median partition P* of a finite set X is a minimizer of a remoteness function φ(P)=∑i=1mD(P,Pi) defined from a profile of partitions E={P1,P2,…,Pm} and a function D that quantifies the distance between two partitions. This paper focuses in the location of median partitions within the lattice of partitions when the symmetric function D is such that, for every partition P, D(P,.) is a submodular function that decreases along any chain (either ascending or descending) starting at P. Especially, refinement relationships among median partitions and quota rules are investigated. Most of the results presented here remain valid in an arbitrary semimodular lattice.

Cartesian Lattice Counting by the Vertical 2-sum

A vertical 2-sum of a two-coatom lattice L and a two-atom lattice U is obtained by removing the top of L and the bottom of U, and identifying the coatoms of L with the atoms of U. This operation creates one or two nonisomorphic lattices depending on the symmetry case. Here the symmetry cases are analyzed, and a recurrence relation is presented that expresses the number of nonisomorphic vertical 2-sums in some desired family of graded lattices. Nonisomorphic, vertically indecomposable modular and distributive lattices are counted and classified up to 35 and 60 elements respectively. Asymptotically their numbers are shown to be at least Ω(2.3122n) and Ω(1.7250n), where n is the number of elements. The number of semimodular lattices is shown to grow faster than any exponential in n.

Characterizations of pseudocomplemented lattices by excluded 0-sublattices

In this paper, we characterize pseudocomplemented lattices with (ACC) and (DCC) by using their sublattices. Specifically speaking, we give a necessary and sufficient condition that a lattice with (ACC) and (DCC) is pseudocomplemented by using its nine sublattices. Moreover, we describe the interval pseudocomplementedness of semimodular lattices with (ACC) and (DCC) by their sublattices equivalently.

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

Following G.~Gratzer and E.~Knapp, 2009, a planar semimodular lattice $L$ is \emph{rectangular}, if~the left boundary chain has exactly one doubly-irreducible element, $c_l$, and the right boundary chain has exactly one doubly-irreducible element, $c_r$, and these elements are complementary. The Czedli-Schmidt Sequences, introduced in 2012, construct rectangular lattices. We use them to prove some structure theorems. In particular, we prove that for a slim (no $\mathsf{M}_3$ sublattice) rectangular lattice~$L$, the congruence lattice $\Con L$ has exactly $\length[c_l,1] + \length[c_r,1]$ dual atoms and a dual atom in $\Con L$ is a congruence with exactly two classes. We also describe the prime ideals in a slim rectangular lattice.

Medians are Below Joins in Semimodular Lattices of Breadth 2

Let L be a lattice of finite length and let d denote the minimum path length metric on the covering graph of L. For any xi =(x_{1},dots ,x_{k})in L^{k}, an element y belonging to L is called a median of ξ if the sum d(y,x1) + ⋯ + d(y,xk) is minimal. The lattice L satisfies the c1-median property if, for any xi =(x_{1},dots ,x_{k})in L^{k} and for any median y of ξ, yleq x_{1}vee dots vee x_{k}. Our main theorem asserts that if L is an upper semimodular lattice of finite length and the breadth of L is less than or equal to 2, then L satisfies the c1-median property. Also, we give a construction that yields semimodular lattices, and we use a particular case of this construction to prove that our theorem is sharp in the sense that 2 cannot be replaced by 3.