Lattice Tensor Products i. Coordinatization

G. Grätzer and F. Wehrung introduced the lattice tensor product, A B, of the lattices A and B. One of the most important properties is that for a simple and bounded lattice A, the lattice A B is a congruence-preserving extension of B. The lattice A B is defined as the set of certain subsets of A B; there is no easy test when a subset belongs to A B. A special case, M ₃B, was earlier defined by G. Gräatzer and F. Wehrung as M ₃, the it Boolean triple construct, defined as a subset of B ³, with a simple criterion when a triple belongs. A~recent paper of G. Grätzer and E. T. Schmidt illustrates the importance of this Boolean triple arithmetic. In this paper we show that for any finite lattice A, we can ``coordinatize" A B, that is, represent A B as a subset of B ⁿ (where n is the number of join-irreducible elements of A), and provide an effective criteria to recognize the n-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we reprove a special case of the above result: for a finite simple lattice A, the lattice A B is a congruence-preserving extension of B.     

Invariants of Composite Networks Arising as a Tensor Product

We generalize Brylawski’s formula of the Tutte polynomial of a tensor product of matroids to colored connected graphs, matroids, and disconnected graphs. Unlike the non-colored tensor product where all edges have to be replaced by the same graph, our colored generalization of the tensor product operation allows individual edge replacement. The colored Tutte polynomials we compute exists by the results of Bollobás and Riordan. The proof depends on finding the correct generalization of the two components of the pointed Tutte polynomial, first studied by Brylawski and Oxley, and on careful enumeration of the connected components in a tensor product. Our results make the calculation of certain invariants of many composite networks easier, provided that the invariants are obtained from the colored Tutte polynomials via substitution and the composite networks are represented as tensor products of colored graphs. In particular, our method can be used to calculate (with relative ease) the expected number of connected components after an accident hits a composite network in which some major links are identical subnetworks in themselves.      

The Spectra of Coxeter Graphs

We determine the spectra of the finite Coxeter graphs defined by a terminal node of the Coxeter diagram, and the spectra of their thick equivalents.     

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs $G$ and $H$. Let $H^k\left[G\right]$ be the lexicographic product of $H^k$ and $G$, where $H^k$ is the lexicographic product of the graph $H$ by itself $k$ times. In this paper, we determine the spectrum of $H^k\left[G\right]$ and $H^k$ when $G$ and $H$ are regular and the Laplacian spectrum of $H^k\left[G\right]$ and $H^k$ for $G$ and $H$ arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10¹⁰⁰ ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.     

区间效应代数的张量积

本文主要研究了区间效应代数的张量积的结构.证明了[0,]与[0,1]的张量积不是[0,1].     

Hochschild Homology of Twisted Tensor Products

We compute the Hochschild homology of some twisted tensor products of algebras, which are a natural generalization of the Ore extensions. We apply our result to the ring D _Q,P(X,/X) of differential operators of the multiparametric affine space, the ring of coordinates of the quantum symplectic 2v-dimensional space and the ring of coordinates of the quantum 2v-dimensional Euclidean space.     

On Tensor Products of l-Operators and bo-Operators

It is proved that the tensor product of two linear operators is a cone summing operator (respectively, order bounded operator) if and only if both operators are cone summing (respectively, order bounded).     

Tensor Products in Categories of Topological Spaces

In this paper symmetric monoidal closed structures on coreflective subcategories of the category of (Hausdorff) topological spaces are studied. We describe all such structures on the category of (Hausdorff) pseudoradial spaces and some of its subcategories and give an example of a coreflective subcategory of the category of Hausdorff topological spaces admitting a proper class of symmetric monoidal closed structures.     

Cayley digraphs and lexicographic product

In this paper, we prove that a Cayley digraph Γ = Cay(G, S) is a nontrivial lexicographical product if and only if there is a nontrivial subgroup H of G such that S∖H is a union of some double cosets of H in G.      

Tensor products of certain modules for the generalized cartan matrix lie algebra

The outer multiplicites of tensor product decompositions are computed for the smallest Euclidean Kac-Moody Lie algebra when one of the tensor factors is a fundamental module or the p-module. Some combinatorial identities are obtained.     

Tensor Products of Non-Archimedean Weighted Spaces of Continuous Functions

It is shown that the completion of the tensor product of two non-Archimedean weighted spaces of continuous functions is topologically isomorphic to another weighted space. Several applications of this result are given.     

Cartesian Closed Topological Categories and Tensor Products

The projective tensor product in a category of topological R-modules (where R is a topological ring) can be defined in Top, the category of topological spaces, by the same universal property used to define the tensor product of R-modules in Set. In this article, we extend this definition to an arbitrary topological category X and study how the Cartesian closedness of X is related to the monoidal closedness of the category of R-module objects in X. Mathematics Subject Classifications (2000) 18D15, 18D35, 18A40.     

Planarity of Cayley graphs of graph products of groups

We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.     

Spectra of Extended Double Cover Graphs

The construction of the extended double cover was introduced by N. Alon [1] in 1986. For a simple graph G with vertex set V = {v ₁, v ₂, ..., v _n }, the extended double cover of G, denoted G ^*, is the bipartite graph with bipartition (X, Y) where X = {x ₁, x ₂, ..., x _n } and Y = {y ₁, y ₂, ..., y _n }, in which x _i and y _j are adjacent iff i = j or v _i and v _j are adjacent in G.In this paper we obtain formulas for the characteristic polynomial and the spectrum of G ^* in terms of the corresponding information of G. Three formulas are derived for the number of spanning trees in G ^* for a connected regular graph G. We show that while the extended double covers of cospectral graphs are cospectral, the converse does not hold. Some results on the spectra of the nth iterared double cover are also presented.     

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov–Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.