Topological Aspects of the Product of Lattices

Let be an arbitrary nonempty set and a lattice of subsets of such that ∅, X∈L. () denotes the algebra generated by , and () denotes those nonnegative, finite, finitely additive measures on (). In addition, () denotes the subset of () which consists of the nontrivial zero-one valued measures. The paper gives detailed analysis of products of lattices, their associated Wallman spaces, and products of a variety of measures

Fremlin tensor product respects the unbounded convergences

Suppose $\Sigma$ is a topological space and $S(\Sigma)$ is the vector lattice of all continuous real-valued functions defined on open dense subsets of $\Sigma$. In this paper, we establish some lattice and topological aspects of $S(\Sigma)$. In particular, as an application, we show that the unbounded order convergence and the order convergence are stable under passing to the Fremlin tensor product of two Archimedean vector lattices. Furthermore, by considering the Fremlin projective tensor product between Banach lattices, we establish a partial variant of this result for unbounded norm convergence and unbounded absolute weak convergence, as well.Comment: 11 pages. Submitte

Noncommutative Lattices and Their Continuum Limits

We consider finite approximations of a topological space $M$ by noncommutative lattices of points. These lattices are structure spaces of noncommutative $C^*$-algebras which in turn approximate the algebra \cc(M) of continuous functions on $M$. We show how to recover the space $M$ and the algebra \cc(M) from a projective system of noncommutative lattices and an inductive system of noncommutative $C^*$-algebras, respectively.Comment: 22 pages, 8 Figures included in the LaTeX Source New version, minor modifications (typos corrected) and a correction in the list of author

A Categorical View on Algebraic Lattices in Formal Concept Analysis

Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a category-theoretical perspective. To this end, we build on the the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory.Comment: 36 page

Lattices and Their Continuum Limits

We address the problem of the continuum limit for a system of Hausdorff lattices (namely lattices of isolated points) approximating a topological space $M$. The correct framework is that of projective systems. The projective limit is a universal space from which $M$ can be recovered as a quotient. We dualize the construction to approximate the algebra ${\cal C}(M)$ of continuous functions on $M$. In a companion paper we shall extend this analysis to systems of noncommutative lattices (non Hausdorff lattices).Comment: 11 pages, 1 Figure included in the LaTeX Source New version, minor modifications (typos corrected) and a correction in the list of author

Topology on the Lattice

We review the method developed in Pisa to determine the topological susceptibility in lattice QCD and present a collection of new and old results obtained by the method.Comment: 10 pages, 7 figures. Contribution to "Sense of Beauty in Physics - a volume in honour of Adriano Di Giacomo" (Pisa University Press, Pisa, 2006), on the occasion of his 70th birthda

Deformed matrix models, supersymmetric lattice twists and N=1/4 supersymmetry

A manifestly supersymmetric nonperturbative matrix regularization for a twisted version of N=(8,8) theory on a curved background (a two-sphere) is constructed. Both continuum and the matrix regularization respect four exact scalar supersymmetries under a twisted version of the supersymmetry algebra. We then discuss a succinct Q=1 deformed matrix model regularization of N=4 SYM in d=4, which is equivalent to a non-commutative $A_4^*$ orbifold lattice formulation. Motivated by recent progress in supersymmetric lattices, we also propose a N=1/4 supersymmetry preserving deformation of N=4 SYM theory on $\R^4$. In this class of N=1/4 theories, both the regularized and continuum theory respect the same set of (scalar) supersymmetry. By using the equivalence of the deformed matrix models with the lattice formulations, we give a very simple physical argument on why the exact lattice supersymmetry must be a subset of scalar subalgebra. This argument disagrees with the recent claims of the link approach, for which we give a new interpretation.Comment: 47 pages, 3 figure