The lattice of quasivarieties of commutative moufang loops

It is proved that a lattice of quasivarieties of an arbitrary variety of commutative Moufang loops either has the power of the continuum or is finite, and that the latter is the case iff is generated by a finite group. It is also stated that the lattice of all quasivarieties of a least nonassociative variety of commutative Moufang loops contains a quasivariety which is generated by a finite quasigroup and has no covers; hence, it has no independent basis of quasi-identities. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 700–720, November–December, 1998.     

Phase transitions for models with a continuum set of spin values on a Bethe lattice

Theoretical and Mathematical Physics - We consider a model with nearest-neighbor interactions and the set $$[0,1]$$ of spin values on a Bethe lattice (Cayley tree) of arbitrary order. This model...     

A characterization of complex lattice homomorphisms

In this paper we discuss the problem of how to recognize a complex lattice homomorphism on the complexification of a real vector lattice L from its behavior on a small subset of .      

On irrational lattice angles

The aim of this paper is to generalize the notions of ordinary and expanded lattice angles and their sums studied in the work  [7] by the author to the case of angles with lattice vertices but not necessarily with lattice rays. We find normal forms and extend the definition of lattice sums to a certain special case of such angles.     

Planar lattice gases with nearest-neighbor exclusion

We discuss the hard-hexagon and hard-square problems, as well as the corresponding problem on the honeycomb lattice. The case when the activity is unity is of interest to combinatorialists, being the problem of counting binary matrices with no two adjacent 1's. For this case, we use the powerful corner transfer matrix method to numerically evaluate the partition function per site, density and some near-neighbor correlations to high accuracy. In particular, for the square lattice, we obtain the partition function per site to 43 decimal places.     

Commutativity of the Arens product in lattice ordered algebras

Let be an Abelian Archimedean lattice ordered algebra. The order bidual furnished with the Arens product is again a lattice ordered algebra. We show that the order continuous order bidual is Abelian. This solves an open problem and improves a result of Scheffold, who proved it for the case of normed lattice ordered algebras. The proof is based on the up-down-up approximation of positive elements in the order continuous order bidual by elements in the canonical image of in Components of positive elements in are characterized and the result is applied to the Arens product of -and almost -algebras.     

Fine structure of the homomorphisms of the lattice of uniformly continuous functions on the line

Positivity - We provide a representation of the homomorphisms $$U\longrightarrow {\mathbb {R}}$$, where U is the lattice of all uniformly continuous functions on the line The resulting picture is...     

The lattice automorphisms of simple algebraic groups over\bar F_2

We prove that if u is a unipotent element of a connected reductive algebraic group G over , there exists an involution σ in G such that σuσ=u⁻¹. We use this result to determine the group of lattice automorphisms of G, when G is simple.     

The ultratriangular form for prime-power lattice rules

A recent development in the theory of lattice rules has been the introduction of the unique ultratriangular D-Z form for prime-power rules. It is known that any lattice rule may be decomposed into its Sylow p-components. These components are prime-power rules, each of which has a unique ultratriangular form. By reassembling these ultratriangular forms in a defined way, it is possible to obtain a canonical form for any lattice rule. A special case occurs when the ultratriangular forms of each of the Sylow p-components have a consistent set of column indices. In this case, it is possible to obtain a unique canonical D-Z form. Given the column indices and the invariants for an ultratriangular form, we may obtain a formula for the number of ultratriangular forms, and hence the number of prime-power lattice rules, having these column indices and invariants.     

Computing on-line the lattice of maximal antichains of posets

We consider the on-line computation of the lattice of maximal antichains of a finite poset . This on-line computation satisfies what we call the linear extension hypothesis: the new incoming vertex is always maximal in the current subposet of . In addition to its theoretical interest, this abstraction of the lattice of antichains of a poset has structural properties which give it interesting practical behavior. In particular, the lattice of maximal antichains may be useful for testing distributed computations, for which purpose the lattice of antichains is already widely used. Our on-line algorithm has a run time complexity of , where |P| is the number of elements of the poset, , |MA(P)| is the number of maximal antichains of and (P) is the width of . This is more efficient than the best off-line algorithms known so far.     

On a relative uniform completion of an archimedean lattice ordered group

The notion of a relatively uniform convergence (ru-convergence) has been used first in vector lattices and then in Archimedean lattice ordered groups. Let G be an Archimedean lattice ordered group. In the present paper, a relative uniform completion (ru-completion) of G is dealt with. It is known that exists and it is uniquely determined up to isomorphisms over G. The ru-completion of a finite direct product and of a completely subdirect product are established. We examine also whether certain properties of G remain valid in . Finally, we are interested in the existence of a greatest convex l-subgroup of G, which is complete with respect to ru-convergence. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

Bounds for the lattice point enumerator

For the lattice point enumerator of a lattice and a convex body K we give bounds in terms of the intrinsic volumes of K and of minimal determinants of . The intrinsic volumes are the normalized Minkowski quermassintegrals and the minimal determinants are analogous functionals of .     

Routing sets in the integer lattice

A set of vertices S in a graph G is a routing set if it ensures some kind of connectivity between all pairs of vertices outside of S. Additional constraints may apply; a connected dominating set, for instance, is a special case of a routing set. We determine the size of a minimum routing set in subgraphs of the integer lattice, as well as (asymptotically) for the lattice itself.     

Convex lattice polygons with all lattice points visible

Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the polygon are visible from it. We completely classify such polygons, show that there are finitely many of lattice width greater than 2, and computationally enumerate them. As an application of this classification, we prove new obstructions to graphs arising as skeleta of tropical plane curves.     

Finite sublattices in the lattice of clones

The lattice of clones of functions over a k-element set is studied. It is shown that every lattice which is a countable direct product of finite lattices is embedded (up to isomorphism) in and, hence, in for k 4. This directly implies that every finite and any countable residually finite lattice is embedded in , k 4, and that no nontrivial quasi-identity holds in , k 4. A number of particular lattices (which are free in some lattice varieties) embeddable in , k 4,are presented. Translated fromAlgebra i Logika, Vol. 33, No. 5, pp. 514–549, September–October, 1994.     

Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

Repleteness and regular lattice measures

Let X be an abstract set and L a lattice of subsets of X. To each lattice regular measure µ, we associate two induced measures and on suitable lattices of the Wallman space IR(L) and another measure µ on the spaceIR(L). We will investigate the reflection of smoothness properties of p onto , and µ; and try to set some new criterion for repleteness and measure repleteness.     

Difference discrete connection and curvature on cubic lattice

In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice. One of the definitions can be extended to the case over the random lattice. We also discuss the relation between our approach and the lattice gauge theory and apply to the discrete integrable systems. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday