Slim Semimodular Lattices. I. A Visual Approach

A finite lattice L is called slim if no three join-irreducible elements of L form an antichain. Slim lattices are planar. After exploring some elementary properties of slim lattices and slim semimodular lattices, we give two visual structure theorems for slim semimodular lattices.     

Slim Semimodular Lattices. II. A Description by Patchwork Systems

Rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in Acta Sci Math 75:29–48, 2009. A patch lattice is a rectangular lattice whose weak corners are coatoms. As a variant of gluing, we introduce the concept of a patchwork system. We prove that every glued sum indecomposable, planar, semimodular lattice is a patchwork of its maximal patch lattice intervals. For a planar modular lattice, our patchwork system is the same as the S-glued system introduced by C. Herrmann in Math Z 130:255–274, 1973. Among planar semimodular lattices, patch lattices are characterized as the patchwork-irreducible ones. They are also characterized as the indecomposable ones with respect to gluing over chains; this gives another structure theorem.     

Quasiplanar Diagrams and Slim Semimodular Lattices

For elements x and y in the (Hasse) diagram D of a finite bounded poset P, x is on the left of y, written as x λ y, if x and y are incomparable and x is on the left of all maximal chains through y. Being on the right, written as x ? y, is defined analogously. The diagram D is quasiplanar if λ and ? are transitive and for any pair (x,y) of incomparable elements, if x is on the left of some maximal chain through y, then x λ y. A planar diagram is quasiplanar, and P has a quasiplanar diagram iff its order dimension is at most 2. We are interested in diagrams only up to similarity. A finite lattice is slim if it is join-generated by the union of two chains. The main result gives a bijection between the set of (the similarity classes of) finite quasiplanar diagrams and that of (the similarity classes of) planar diagrams of finite slim semimodular lattices. This bijection allows one to describe finite posets of order dimension at most 2 by finite slim semimodular lattices, and conversely. As a corollary, we obtain that there are exactly (n?2)! quasiplanar diagrams of size n.     

Semimodular Lattices and Semibuildings

In a ranked lattice, we consider two maximal chains, or flags to be i-adjacent if they are equal except possibly on rank i. Thus, a finite rank lattice is a chamber system. If the lattice is semimodular, as noted in [9], there is a Jordan-Hölder permutation between any two flags. This permutation has the properties of an S_n-distance function on the chamber system of flags. Using these notions, we define a W-semibuilding as a chamber system with certain additional properties similar to properties Tits used to characterize buildings. We show that finite rank semimodular lattices form an S_n-semibuilding, and develop a flag-based axiomatization of semimodular lattices. We refine these properties to axiomatize geometric, modular and distributive lattices as well, and to reprove Tits' result that S_n-buildings correspond to relatively complemented modular lattices (see [16], Section 6.1.5).     

A Characterization of Ordered Sets and Lattices via Betweenness Relations

For a set X with at least 3 elements, we establish a canonical one to one correspondence between all betweenness relations satisfying certain axioms and all pairs of inverse orderings “<” and “>” defined on X for which the corresponding Hasse diagram is connected and all maximal chains contain at least 3 elements. For an ordering “<”, the corresponding betweenness relation B is given by $$B=\{(x,y,z)\in X^3\mid x<y<z {\rm \ or\ }z<y<x\}.$$ Moreover, by adding one more axiom, we obtain also a one to one correspondence between all pairs of dual lattices and all betweenness relations.     

The Asymptotic Number of Planar,Slim, Semimodular Lattice Diagrams

A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. We prove that there exists a positive constant C such that, up to similarity, the number of planar diagrams of slim semimodular lattices of size n is asymptotically C · 2 ⁿ .     

Relations Between Some Dimensions of Semimodular Lattices

The aim of this paper is to present relations between Goldie, hollow and Kurosh-Ore dimensions of semimodular lattices. Relations between Goldie and Kurosh-Ore dimensions of modular lattices were studied by Grzeszczuk, Okiski and Puczyowski.     

Varieties of Lattices with Geometric Descriptions

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann et al. proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem for free lattices. This solves two problems stated by Huhn in 1985. We prove that every modular (resp., n-distributive) lattice embeds within its variety into some strongly spatial lattice. Every lattice which is either algebraic modular spatial or bi-algebraic is strongly spatial. We also construct a lattice that cannot be embedded, within its variety, into any algebraic and spatial lattice. This lattice has a least and a largest element, and it generates a locally finite variety of join-semidistributive lattices.     

Two Notes on the Variety Generated by Planar Modular Lattices

Let Var(M _plan) denote the variety generated by the class M _plan of planar modular lattices. In 1977, based on his structural investigations, R. Freese proved that Var(M _plan) has continuumly many subvarieties. The present paper provides a new approach to this result utilizing lattice identities. We also show that each subvariety of Var(M _plan) is generated by its planar (subdirectly irreducible) members. Dedicated to the memory of András P. Huhn This research was partially supported by the NFSR of Hungary (OTKA), grant no. T 049433, T 48809 and K 60148.     

Graph minors. III. Planar tree-width

The “tree-width” of a graph is defined and it is proved that for any fixed planar graph H, every planar graph with sufficiently large tree-width has a minor isomorphic to H. This result has several applications which are described in other papers in this series.     

Lower Semimodular Types of Lattices: Frankl's Conjecture Holds for Lower Quasi-Semimodular Lattices

We introduce a measure of how far a lattice L is from being lower semimodular. We call it the lower semimodular type of L. A lattice has lower semimodular type zero if and only if it is lower semimodular. In this paper we discuss properties of the measure and we show that Frankl's conjecture holds for lower quasi-semimodular lattices: if a lattice L is lower quasi-semimodular then there is a join-irreducible element x in L such that the size of the principal filter generated by x is at most (|L|− 1) /2. Revised: July 2, 1997     

Large Sets of Lattices without Order Embeddings

Let I and μ be an infinite index set and a cardinal, respectively, such that |I| ≤ μ and, starting from ?₀, μ can be constructed in countably many steps by passing from a cardinal λ to 2^λ at successor ordinals and forming suprema at limit ordinals. We prove that there exists a system X = {L_i: i ∈ I} of complemented lattices of cardinalities less than |I| such that if i, j ∈ I and φ: L_i → L_j is an order embedding, then i = j and φ is the identity map of L_i. If |I| is countable, then, in addition, X consists of finite lattices of length 10. Stating the main result in other words, we prove that the category of (complemented) lattices with order embeddings has a discrete full subcategory with |I| many objects. Still in other words, the class of these lattices has large antichains (that is, antichains of size |I|) with respect to the quasiorder “embeddability.” As corollaries, we trivially obtain analogous statements for partially ordered sets and semilattices.     

Isometrical Embeddings of Lattices into Geometric Lattices

A lattice is said to be finite height generated if it is complete and every element is the join of some elements of finite height. Extending former results by Grätzer and Kiss (Order 2:351–365, 1986) on finite lattices, we prove that every finite height generated algebraic lattice that has a pseudorank function is isometrically embeddable into a geometric lattice.     

Minimal Paths between Maximal Chains in Finite Rank Semimodular Lattices

We study paths between maximal chains, or flags, in finite rank semimodular lattices. Two flags are adjacent if they differ on at most one rank. A path is a sequence of flags in which consecutive flags are adjacent. We study the union of all flags on at least one minimum length path connecting two flags in the lattice. This is a subposet of the original lattice. If the lattice is modular, the subposet is equal to the sublattice generated by the flags. It is a distributive lattice which is determined by the Jordan-Hölder permutation between the flags. The minimal paths correspond to all reduced decompositions of this permutation. In a semimodular lattice, the subposet is not uniquely determined by the Jordan-Hölder permutation for the flags. However, it is a join sublattice of the distributive lattice corresponding to this permutation. It is semimodular, unlike the lattice generated by the two flags, which may not be ranked. The minimal paths correspond to some reduced decompositions of the permutation, though not necessarily all. We classify the possible lattices which can arise in this way, and characterize all possibilities for the set of shortest paths between two flags in a semimodular lattice.     

Semimodular lattices and the Hall-Dilworth gluing construction

We present a new gluing construction for semimodular lattices, related to the Hall-Dilworth construction     

Counting Finite Lattices

The correct values for the number of all unlabeled lattices on n elements are known for . We present a fast orderly algorithm generating all unlabeled lattices up to a given size n. Using this algorithm, we have computed the number of all unlabeled lattices as well as that of all labeled lattices on an n-element set for each . Received April 4, 2000; accepted in final form November 2, 2001. RID="h1" ID="h1" Presented by R. Freese.     

Planar lattices and planar graphs

It is shown that a finite lattice is planar if and only if the (undirected) graph obtained from its (Hasse) diagram by adding an edge between its least and greatest elements is a planar graph.     

Frankl's Conjecture Is True for Lower Semimodular Lattices

It is shown that every finite lower semimodular lattice L with |L|≥2 contains a join-irreducible element x such that at most |L|/2 elements y∈L satisfy y≥x. Revised: August 16, 1999     

拟紧元和连续格

在连续格中, 引入了拟紧元和拟基的概念, 在研究了它们的基本性质的基础上, 给出了连续格的一种表示定理.     

The Matrix of a Slim Semimodular Lattice

A finite lattice L is called slim if no three join-irreducible elements of L form an antichain. Slim semimodular lattices play the main role in G. Czédli and E. T. Schmidt (Algebra Univers, to appear), where lattice theory is applied to a purely group theoretical problem. Here we develop a unique matrix representation for these lattices.