Iterations of Boolean algebras with measure

We consider a classM of Boolean algebras with strictly positive, finitely additive measures. It is shown thatM is closed under iterations with finite support and that the forcing via such an algebra does not destroy the Lebesgue measure structure from the ground model. Also, we deduce a simple characterization of Martin's Axiom reduced to the classM.     

On multiplicity theory for Boolean algebras of projections

The aim of this paper is to present a study of the connections between the commutants of a Boolean algebra of projections of finite multiplicity and the uniformly closed algebra generated by these projections. This paper is a part of the author’s Ph.D. thesis to be submitted to the Hebrew University. The author wishes to express his thanks to Professor S. R. Foguel for much valuable advice and encouragement.     

Boolean Metric Spaces and Boolean Algebraic Varieties

Abstract

The concepts of Boolean metric space and convex combination are used to characterize polynomial maps A ⁿ ?→?A ^m in a class of commutative Von Neumann regular rings including p-rings, that we have called CFG-rings. In those rings, the study of the category of algebraic varieties (i.e., sets of solutions to a finite number of polynomial equations with polynomial maps as morphisms) is equivalent to the study of a class of Boolean metric spaces, that we call here CFG-spaces.     

Skew Boolean algebras derived from generalized Boolean algebras

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/ where is the Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras constructed from generalized Boolean algebras B by a twisted product construction for which . In particular we study the congruence lattice of with an eye to viewing as a minimal skew Boolean cover of B. This construction is the object part of a functor from the category GB of generalized Boolean algebras to the category LSB of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor . This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.     

Pseudo-trees and Boolean algebras

We consider Boolean algebras constructed from pseudo-trees in various ways and make comments about related classes of Boolean algebras.     

Fuzzy Boolean algebras

The purpose of this paper is to generalize the following situation: from the concrete structure $\text{B}$, we define the notion of Boolean algebras; the Stone representation theorem allows us to replace the algebraic study of Boolean algebras by a topological one. Let E be a non-empty set, and J a non-empty ordered set. Note $\text{B}$ the set of all fuzzy subsets of (E,J). We shall introduce the concept of fuzzy Boolean algebra and find a representation theorem. But it will be difficult to speak of the dual fuzzy topological space of a fuzzy Boolean algebra as we shall see further, except in certain particular cases.     

A nonstandard proof of a lemma from constructive measure theory

Suppose that f_n is a sequence of nonnegative functions with compact support on a locally compact metric space, that T is a nonnegative linear functional, and that T f_n < T f₀. A result of Bishop, foundational to a constructive theory of functional analysis, asserts the existence of a point x such that f_n (x) < f ₀(x). This paper extends this result to arbitrary Hausdorff spaces, and gives short proofs using nonstandard analysis. While such arguments used are not themselves constructive, they can illuminate where the difficulty lies in finding the point x. An algorithm for constructing x is then given, with a nonstandard proof that the algorithm converges to a correct value. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boolean algebras R-generated by MV-effect algebras

We prove that every MV-effect algebra M is, as an effect algebra, a homomorphic image of its R-generated Boolean algebra. We characterize central elements of M in terms of the constructed homomorphism.     

Principal functions,index theory,geometric measure theory and function algebras

The aim of this paper is twofold: first to provide evidence of a nice relationship between index theory and operator algebras within the framework of geometric measure theory by exhibiting basic examples involving one dimensional singular integral operators; second to expose certain connections that exist involving the principal function associated to an operator having trace class self-commutator and the theory of function algebras.     

Unsupported Boolean algebras and forcing

If κ is an infinite cardinal, a complete Boolean algebra B is called κ‐supported if for each sequence 〈b_β : β < κ〉 of elements of B the equality _{α<κ β>α} b_β = equation/tex2gif-inf-5.gif _β∈A b_β holds. Combinatorial and forcing equivalents of this property are given and compared with the other forcing related properties of Boolean algebras (distributivity, caliber, etc.). The set of regular cardinals κ for which B is not κ‐supported is investigated. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)