Congruence-preserving extensions of finite lattices to sectionally complemented lattices

In 1962, the authors proved that every finite distributive lattice can be represented as the congruence lattice of a finite sectionally complemented lattice. In 1992, M. Tischendorf verified that every finite lattice has a congruence-preserving extension to an atomistic lattice. In this paper, we bring these two results together. We prove that every finite lattice has a congruence-preserving extension to a finite sectionally complemented lattice.

     

Comparability graphs of lattices

A theorem of N. Terai and T. Hibi for finite distributive lattices and a theorem of Hibi for finite modular lattices (suggested by R.P. Stanley) are equivalent to the following: if a finite distributive or modular lattice of rank d contains a complemented rank 3 interval, then the lattice is (d+1)-connected.In this paper, the following generalization is proved: Let L be a (finite or infinite) semimodular lattice of rank d that is not a chain (d∈N₀). Then the comparability graph of L is (d+1)-connected if and only if L has no simplicial elements, where z∈L is simplicial if the elements comparable to z form a chain.     

Sublattices of associahedra and permutohedra

Grätzer asked in 1971 for a characterization of sublattices of Tamari lattices. A natural candidate was coined by McKenzie in 1972 with the notion of a bounded homomorphic image of a free lattice—in short, bounded lattice. Urquhart proved in 1978 that every Tamari lattice is bounded (thus so are its sublattices). Geyer conjectured in 1994 that every finite bounded lattice embeds into some Tamari lattice.     

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.     

A Ramsey-type theorem for traceable graphs

A graph G is traceable if there is a path passing through all the vertices of G. It is proved that every infinite traceable graph either contains arbitrarily large finite chordless paths, or contains a subgraph isomorphic to graph A, illustrated in the text. A corollary is that every finitely generated infinite lattice of length 3 contains arbitrarily large finite fences. It is also proved that every infinite traceable graph containing no chordless four-point path contains a subgraph isomorphic to K_ω,ω. The versions of these results for finite graphs are discussed.     

Lattice Classification by Cut-through Coding

Inspired by engineering of high-speed switching with quality of service, this paper introduces a new approach to classify finite lattices by the concept of cut-through coding. An n-ary cut-through code of a finite lattice encodes all lattice elements by distinct n-ary strings of a uniform length such that for all j, the initial j encoding symbols of any two elements x and y determine the initial j encoding symbols of the meet and join of x and y. In terms of lattice congruences, some basic criteria are derived to characterize the n-ary cut-through codability of a finite lattice. N-ary cut-through codability also gives rise to a new classification of lattice varieties and in particular, defines a chain of ideals in the lattice of lattice varieties.     

Sums of sets of lattice points and unimodular coverings of polytopes

If P is a lattice polytope (that is, the convex hull of a finite set of lattice points in \({\mathbf{R}^n}\)), then every sum of h lattice points in P is a lattice point in the h-fold sumset hP. However, a lattice point in the h-fold sumset hP is not necessarily the sum of h lattice points in P. It is proved that if the polytope P is a union of unimodular simplices, then every lattice point in the h-fold sumset hP is the sum of h lattice points in P.     

Finite Lattices as Lattices of Relative Congruences of Finite Unars and Abelian Groups

It is proved that every finite lattice is isomorphic to an R-congruence lattice of a finite unar (finite Abelian group), as well as to a lattice of R-varieties for some locally finite finitely axiomatizable quasivariety of unars (Abelian groups) R.     

Minimality of horospherical flows

IfN is the nilpotent constituent of an Iwasawa decomposition of the semi-simple groupG (finite center and no compact factors), it is proved thatN acts minimally onG/Γ for every uniform lattice Γ ?G, generalizing theorems of Hedlund and L. Greenberg.     

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.     

Notes on sectionally complemented lattices. I

Summary In 1944, R.P. Dilworth proved (unpublished) that every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L. In 1960, G. Grätzer and E. T. Schmidt improved this result by constructing a finite sectionally complemented lattice L whose congruence lattice represents D. In L, sectional complements do not have to be unique. The one sectional complement constructed by G. Grätzer and E. T. Schmidt in 1960, we shall call the 1960 sectional complement. This paper examines it in detail. The main result is an algebraic characterization of the 1960 sectional complement.     

Generalized Frattini subgroups as coradicals of groups

The paper deals with finite solvable groups only. It is established that the class of all regular subgroup m-functors coincides with the class of all X-abnormal m-functors, where X ranges over all subclasses of the class of all primitive groups. The properties of the lattice of all regular subgroup m-functors are studied and the atoms and coatoms of this lattice are described. It is proved that the generalized Frattini subgroup of G corresponding to a regularm-functor coincides with the X-coradical of G for some R ₀-closed class X.     

Lattice polytopes of degree 2

A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly i>0 interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimensions. In particular, there is only a finite number of quadratic polynomials with fixed leading coefficient being the h^∗-polynomial of a lattice polytope.     

Embedding Ordered Sets into Distributive Lattices

This paper investigates the class of ordered sets that are embeddable into a distributive lattice in such a way that all existing finite meets and joins are preserved. The main result is that the following decision problem is NP-complete: Given a finite ordered set, is it embeddable into a distributive lattice with preservation of existing meets and joins? The NP-hardness of the problem is proved by polynomial reduction of the classical 3SAT decision problem into it, and the NP-completeness by presenting a suitable NP-algorithm.     

Lattice tensor products. IV

In 1999, for lattices A and B, G. Grätzer and F. Wehrung introduced the lattice tensor product, A?B. In Part I of this paper, we showed that for a finite lattice A and a bounded lattice B, this construction can be "coordinatized,'' that is, represented in B ^A so that the representing elements are easy to recognize. In this note, we show how to extend our method to an arbitrary bounded lattice A to coordinatize A?B.     

Finite elements in some vector lattices of nonlinear operators

We study the collection of finite elements \(\Phi _{1}\big ({\mathcal {U}}(E,F)\big )\) in the vector lattice \({\mathcal {U}}(E,F)\) of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F, where F is Dedekind complete. In particular, for an atomic vector lattice E it is proved that for a finite element in \(\varphi \in {\mathcal {U}}(E,{\mathbb {R}})\) there is only a finite set of mutually disjoint atoms, where \(\varphi \) does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of \(\sigma \)-laterally continuous abstract Uryson functionals. We also describe the ideal \(\Phi _{1}\big ({\mathcal {U}}({\mathbb {R}}^n,{\mathbb {R}}^m)\big )\) for \(n,m\in {\mathbb {N}}\) and consider rank one operators to be finite elements in \({\mathcal {U}}(E,F)\).     

Upper Bounds in Affine Weyl Groups under the Weak Order

Björner and Wachs proved that under the weak order every quotient of a Coxeter group is a meet semi-lattice, and in the finite case is a lattice. In this paper, we examine the case of an affine Weyl group W with corresponding finite Weyl group W ₀. In particular, we show that the quotient of W by W ₀ is a lattice and that up to isomorphism this is the only quotient of W which is a lattice. We also determine that the question of which pairs of elements of W have upper bounds can be reduced to the analogous question within a particular finite subposet.     

On the lattice automorphisms of the finite Chevalley groups

A lattice automorphism of a group is defined to be an automorphism of its lattice of subgroups. For a large class of finite simple Chevalley groups, it is shown that every lattice automorphism is induced by a group automorphism. However, this does not hold for all finite simple Chevalley groups G, as is shown by explicit construction in the case G=PSL(3, q).