Semimodular Lattices Research Articles

Notes on planar semimodular lattices. VIII. Congruence lattices of SPS lattices

In this note, I find a new property of the congruence lattice, $${{\,\mathrm{Con}\,}}L$$, of an SPS lattice L (slim, planar, semimodular, where “slim” is the absence of $${\mathsf {M}}_3$$ sublattices) with more than 2 elements: there are at least two dual atoms in$${{\,\mathrm{Con}\,}}L$$. So the three-element chain cannot be represented as the congruence lattice of an SPS lattice, supplementing a result of G. Czédli.

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.

Frankl's Conjecture for a subclass of semimodular lattices

In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices having breadth at most two. We provide a very short proof of the Conjecture for the class of lattices having breadth at most two. This generalizes the results of Joshi, Waphare and Kavishwar as well as Czedli and Schmidt.

A convex combinatorial property of compact sets in the plane and its roots in lattice theory

K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ for the more general case where $,mathcal U_0$ and $,mathcal U_1$ are compact sets in the plane such that $,mathcal U_1$ is obtained from $,mathcal U_0$ by a positive homothety or by a translation. Also, we give a short survey to show how lattice theoretical antecedents, including a series of papers on planar semimodular lattices by G. Gratzer and E. Knapp, lead to our result.

Uniform semimodular lattices and valuated matroids

In this paper, we present a lattice-theoretic characterization for valuated matroids, which is an extension of the well-known cryptomorphic equivalence between matroids and geometric lattices (= atomistic semimodular lattices). We introduce a class of semimodular lattices, called uniform semimodular lattices, and establish a cryptomorphic equivalence between integer-valued valuated matroids and uniform semimodular lattices. Our result includes a coordinate-free lattice-theoretic characterization of integer points in tropical linear spaces, incorporates the Dress-Terhalle completion process of valuated matroids, and establishes a smooth connection with Euclidean buildings of type A.

Exponential lower bounds of lattice counts by vertical sum and 2-sum

We consider the problem of finding lower bounds on the number of unlabeled n-element lattices in some lattice family. We show that if the family is closed under vertical sum, exponential lower bounds can be obtained from vertical sums of small lattices whose numbers are known. We demonstrate this approach by establishing that the number of modular lattices is at least 2.2726^n for n large enough. We also present an analogous method for finding lower bounds on the number of vertically indecomposable lattices in some family. For this purpose we define a new kind of sum, the vertical 2-sum, which combines lattices at two common elements. As an application we prove that the numbers of vertically indecomposable modular and semimodular lattices are at least 2.1562^n and 2.6797^n for n large enough.

An extension of F. Šik’s theorem on modular lattices

Abstract J. Jakubík noted in [JAKUBÍK, J.: Modular Lattice of Locally Finite Length, Acta Sci. Math. 37 (1975), 79–82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving sublattices isomorphic to S 7. In the present paper we extend the scope of Šik’s theorem to the class of upper semimodular, upper continuous and strongly atomic lattices. Moreover, we show that corresponding result of Jakubík from [JAKUBÍK, J.: Modular Lattice of Locally Finite Length, Acta Sci. Math. 37 (1975), 79–82] cannot be strengthened is analogous way.

Semimodularity and the Jordan–Hölder theorem in posets, with applications to partial partitions

Lattice-theoretical generalizations of the Jordan–Holder theorem of group theory give isomorphisms between finite maximal chains with same endpoints. The best one has been given by Czedli and Schmidt (after Gratzer and Nation), and it applies to semimodular lattices and gives a chain isomorphism by iterating up and down the perspectivity relation between intervals $$[x\wedge y,x]$$ and $$[y,x\vee y]$$ where x covers $$x\wedge y$$ and $$x\vee y$$ covers y. In this paper, we extend to arbitrary (and possibly infinite) posets the definitions of standard semimodularity and of the slightly weaker “Birkhoff condition”, following the approach of Ore (Bull Amer Math Soc 49(8):567–568, 1943). Instead of perspectivity, we associate tags to the covering relation, a more flexible approach. We study the finiteness and length constancy of maximal chains under both conditions and obtain Jordan–Holder theorems. Our theory is easily applied to groups, to closure ranges of an arbitrary poset, and also to five new order relations on the set of partial partitions of a set (i.e. partitions of its subsets), which do not constitute lattices.

Generating Modular Lattices of up to 30 Elements

An algorithm is presented for generating finite modular, semimodular, graded, and geometric lattices up to isomorphism. Isomorphic copies are avoided using a combination of the general-purpose graph-isomorphism tool nauty and some optimizations that handle simple cases directly. For modular and semimodular lattices, the algorithm prunes the search tree much earlier than the method of Jipsen and Lawless, leading to a speedup of several orders of magnitude. With this new algorithm modular lattices are counted up to 30 elements, semimodular lattices up to 25 elements, graded lattices up to 21 elements, and geometric lattices up to 34 elements. Some statistics are also provided on the typical shape of small lattices of these types.

Congruence structure of planar semimodular lattices: the General Swing Lemma

The Swing Lemma, proved by G. Gratzer in 2015, describes how a congruence spreads from a prime interval to another in a slim (having no $$\mathsf {M}_{3}$$ sublattice), planar, semimodular lattice. We generalize the Swing Lemma to planar semimodular lattices.

Congruences and trajectories in planar semimodular lattices

A 1955 result of J. Jakubik states that for the prime intervals p and q of a finite lattice, con(p) ≥ con(q) iff p is congruence-projective to q (via intervals of arbitrary size). The problem is how to determine whether con(p) ≥ con(q) involving only prime intervals. Two recent papers approached this problem in different ways. G. Czedli’s used trajectories for slim rectangular lattices—a special subclass of slim, planar, semimodular lattices. I used the concept of prime-projectivity for arbitrary finite lattices. In this note I show how my approach can be used to reprove Czedli’s result and generalize it to arbitrary slim, planar, semimodular lattices.

A Characterization of a Semimodular Lattice

A geometric lattice is the lattice of closed subsets of a closure operator on a set which is zero-closure, algebraic, atomistic and which has the so-called exchange property. There are many profound results about this type of lattices, the most recent one of which, due to Czedli and Schimdt (Adv Math 225:2455–2463, 2010), says that a lattice L of finite length is semimodular if and only if L has a cover-preserving embedding into a geometric lattice G of the same length. The goal of our paper is to offer the following result: a lattice of finite length is semimodular if and only if every cell in L is a 4-element Boolean lattice and the 7-element non-distributive atomistic lattice having 3 atoms is not a cover-preserving sublattice of L.

Frankl's Conjecture for Subgroup Lattices

We show that the subgroup lattice of any finite group satisfies Frankl’s Union-Closed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common technical result used to prove both may be of some independent interest.

Decompositions in Complete Lattices II. Replaceable Irredundant Decompositions

A characterization of lattices with replaceable irredundant decompositions is given in the following six classes: the class of upper and lower continuous lattices; the class of upper continuous completely join-semidistributive lattices; the class of upper semimodular lower continuous lattices; the class of upper semimodular completely joinsemidistributive lattices; the class of consistent lower continuous lattices; the class of consistent completely join-semidistributive lattices.

Swing Lattice Game and a direct proof of the Swing Lemma for planar semimodular lattices

Swing Lattice Game and a direct proof of the Swing Lemma for planar semimodular lattices

Diagrams and rectangular extensions of planar semimodular lattices

In 2009, G. Gratzer and E. Knapp proved that every planar semimodular lattice has a rectangular extension. We prove that, under reasonable additional conditions, this extension is unique. This theorem naturally leads to a hierarchy of special diagrams of planar semimodular lattices. These diagrams are unique in a strong sense; we also explore many of their additional properties. We demonstrate the power of our new classes of diagrams in two ways. First, we prove a simplified version of our earlier Trajectory Coloring Theorem, which describes the inclusion con\({(\mathfrak{p}) \supseteq}\) con\({(\mathfrak{q})}\) for prime intervals \({\mathfrak{p}}\) and \({\mathfrak{q}}\) in slim rectangular lattices. Second, we prove G. Gratzer’s Swing Lemma for the same class of lattices, which describes the same inclusion more simply.

A proof of Frankl’s union-closed sets conjecture for dismantlable lattices

In this paper, we prove Frankl’s Union-Closed Sets Conjecture for the class of dismantlable lattices, a more general class than the class of planar lattices. As a consequence of this result, we also prove that an upper semimodular lattice with breadth at most two satisfies the Conjecture.

Fast Möbius Inversion in Semimodular Lattices and ER-labelable Posets

We consider the problem of fast zeta and Möbius transforms in finite posets, particularly in lattices. It has previously been shown that for a certain family of lattices, zeta and Möbius transforms can be computed in $O(e)$ elementary arithmetic operations, where $e$ denotes the size of the covering relation. We show that this family is exactly that of geometric lattices. We also extend the algorithms so that they work in $e$ operations for all semimodular lattices, including chains and divisor lattices. Finally, for both transforms, we provide a more general algorithm that works in $e$ operations for all ER-labelable posets.

Congruences of fork extensions of slim, planar, semimodular lattices

For a slim, planar, semimodular lattice L and a covering square S of L, G. Czédli and E. T. Schmidt introduced the fork extension, L[S], which is also a slim, planar, semimodular lattice. This paper investigates when a congruence of L extends to L[S]. We introduce a join-irreducible congruence $${\gamma}$$ (S) of L[S] and determine when it is new, in the sense that it is not generated by a join-irreducible congruence of L. We provide a complete description of $${\gamma}$$ (S). Then we prove that $${\gamma}$$ (S) has at most two covers in the order of join-irreducible congruences of L[S]. Finally, we derive the main result of this paper, the Two-cover Theorem: Every join-irreducible congruence has at most two covers in the order of join-irreducible congruences of a slim, planar, semimodular lattice L.

On the number of slim, semimodular lattices

AbstractA latticeLisslimif it is finite and the set of its join-irreducible elements contains no three-element antichain. Slim, semimodular lattices were previously characterized by G. Czédli and E. T. Schmidt as the duals of the lattices consisting of the intersections of the members of two composition series in a group. Our main result determines the number of (isomorphism classes of) these lattices of a given size in a recursive way. The corresponding planar Hasse diagrams, up to similarity, are also enumerated. We prove that the number of diagrams of slim, distributive lattices of a given lengthnis thenth Catalan number. Besides lattice theory, the paper includes some combinatorial arguments on permutations and their inversions.