Hyper‐Archimedean BL‐algebras are MV‐algebras

Generalizations of Boolean elements of a BL‐algebra L are studied. By utilizing the MV‐center MV(L) of L, it is reproved that an element x ∈ L is Boolean iff x ∨ x * = 1 . L is called semi‐Boolean if for all x ∈ L, x * is Boolean. An MV‐algebra L is semi‐Boolean iff L is a Boolean algebra. A BL‐algebra L is semi‐Boolean iff L is an SBL‐algebra. A BL‐algebra L is called hyper‐Archimedean if for all x ∈ L, xⁿ is Boolean for some finite n ≥ 1. It is proved that hyper‐Archimedean BL‐algebras are MV‐algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV‐algebras or BL‐algebras. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A topology induced by uniformity on BL‐algebras

In this paper, we consider a collection of filters of a BL‐algebra A. We use the concept of congruence relation with respect to filters to construct a uniformity which induces a topology on A. We study the properties of this topology regarding different filters. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the nuclearity of certain Cuntz‐Pimsner algebras

In the present paper, we give a short proof of the nuclearity property of a class of Cuntz‐Pimsner algebras associated with a Hilbert ??‐bimodule ??, where ?? is a separable and nuclear C *‐algebra. We assume that the left ??‐action on the bimodule ?? is given in terms of compact module operators and that ?? is direct summand of the standard Hilbert module over ?? (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boolean products of R0‐algebras

In this paper, the (weak) Boolean representation of R₀‐algebras are investigated. In particular, we show that directly indecomposable R₀‐algebras are equivalent to local R₀‐algebras and any nontrivial R₀‐algebra is representable as a weak Boolean product of local R₀‐algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Representations of MV‐algebras by sheaves

In this paper, inspired by methods of Bigard, Keimel, and Wolfenstein ([2]), we develop an approach to sheaf representations of MV‐algebras which combines two techniques for the representation of MV‐algebras devised by Filipoiu and Georgescu ([18]) and by Dubuc and Poveda ([16]). Following Davey approach ([12]), we use a subdirect representation of MV‐algebras that is based on local MV‐algebras. This allowed us to obtain: (a) a representation of any MV‐algebras as MV‐algebra of all global sections of a sheaf of local MV‐algebras on the spectruum of its prime ideals; (b) a representation of MV‐algebras, having the space of minimal prime ideals compact, as MV‐algebra of all global sections of a Hausdorff sheaf of MV‐chains on the space of minimal prime ideals, which is a Stone space; (c) an adjunction between the category of all MV‐algebras and the category of MV‐algebraic spaces, where an MV‐algebraic space is a pair (X, F), where X is a compact topological space and F is a sheaf of MV‐algebras with stalks that are local (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

n‐linear weakly Heyting algebras

The present paper introduces and studies the variety ????_n of n‐linear weakly Heyting algebras. It corresponds to the algebraic semantic of the strict implication fragment of the normal modal logic K with a generalization of the axiom that defines the linear intuitionistic logic or Dummett logic. Special attention is given to the variety ????₂ that generalizes the linear Heyting algebras studied in [10] and [12], and the linear Basic algebras introduced in [2]. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A solution to identities problem in 2‐element HSI‐algebras

All arithmetical identities involving 1, addition, multiplication and exponentiation are valid in every 2‐element HSI‐algebra. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the category of hyper MV‐algebras

In this paper we study the category of hyper MV‐algebras and we prove that it has a terminal object and a coequalizer. We show that Jia's construction can be modified to provide a free hyper MV‐algebra by a set. We use this to show that in the category of hyper MV‐algebras the monomorphisms are exactly the one‐to‐one homomorphisms. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On many‐sorted algebraic closure operators

A theorem of Birkhoff‐Frink asserts that every algebraic closure operator on an ordinary set arises, from some algebraic structure on the set, as the corresponding generated subalgebra operator. However, for many‐sorted sets, i.e., indexed families of sets, such a theorem is not longer true without qualification. We characterize the corresponding many‐sorted closure operators as precisely the uniform algebraic operators. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weighted composition operators and locally convex algebras

The Gleason-Kahane-Zelazko theorem characterizes the continuous homo-morphism of an associative, locally multiplicatively convex, sequentially complete algebra A into the field C among all linear forms on A. This characterization will be applied along two different directions. In the case in which A is a commutative Banach algebra, the theorem yields the representation of some classes of continuous linear maps A:A→A as weighted composition operators, or as composition operators when A is a continuous algebra endomorphism. The theorem will then be applied to explore the behaviour of continuous linear forms on quasi-regular elements, when A is either the algebra of all Hilbert-Schmidt operators or a Hilbert algebra.     

n ‐fold filters in BL‐algebras

In this paper we introduce n ‐fold (positive) implicative basis logic and the related algebras called n ‐fold (positive) implicative BL‐algebras. Also we define n ‐fold (positive) implicative filters and we prove some relations between these filters and construct quotient algebras via these filters. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Radical of filters in BL‐algebras

In this paper, the notion of the radical of a filter in BL‐algebras is defined and several characterizations of the radical of a filter are given. Also we prove that A/F is an MV‐algebra if and only if D_s(A) ? F. After that we define the notion of semi maximal filter in BL‐algebras and we state and prove some theorems which determine the relationship between this notion and the other types of filters of a BL‐algebra. Moreover, we prove that A/F is a semi simple BL‐algebra if and only if F is a semi maximal filter of A. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Mellin‐edge representations of elliptic operators

We construct a class of elliptic operators in the edge algebra on a manifold M with an embedded submanifold Y interpreted as an edge. The ellipticity refers to a principal symbolic structure consisting of the standard interior symbol and an operator‐valued edge symbol. Given a differential operator A on M for every (sufficiently large) s we construct an associated operator ??_s in the edge calculus. We show that ellipticity of A in the usual sense entails ellipticity of ??_s as an edge operator (up to a discrete set of reals s). Parametrices P of A then correspond to parametrices ??_s of ??_s, interpreted as Mellin‐edge representations of P. Copyright © 2005 John Wiley & Sons, Ltd.     

Some results in BL ‐algebras

We define the set of double complemented elements in BL‐algebras and state and prove some theorems which determines properties of these sets. We introduce the notion of an almost top element and study the properties of these elements (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Amenable Köthe co‐echelon algebras

In this note we investigate amenability properties in the class of the so‐called DF‐algebras. We also characterize, in terms of the defining sequence of weights, amenable Köthe co‐echelon algebras.     

The Banach algebras generated by operators with one-point spectrum

In this article, we present some results related to the structure of Banach algebras generated by operators with one-point spectrum.     

Compact operators and Toeplitz algebras on multiply-connected domains

If Ω is a smoothly bounded multiply-connected domain in the complex plane and S belongs to the Toeplitz algebra τ of the Bergman space of Ω, we show that S is compact if and only if its Berezin transform vanishes at the boundary of Ω. We also show that every element S in T, the C^?-subalgebra of τ generated by Toeplitz operators with symbols in H^∞(Ω), has a canonical decomposition for some R in the commutator ideal CT; and S is in CT iff the Berezin transform vanishes identically on the set M₁ of trivial Gleason parts.     

A discrete representation of free MV‐algebras

We prove that the m ‐generated free MV‐algebra is isomorphic to a quotient of the disjoint union of all the m ‐generated free MV⁽ⁿ⁾‐algebras. Such a quotient can be seen as the direct limit of a system consisting of all free MV⁽ⁿ⁾‐algebras and special maps between them as morphisms (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classification of Rota-Baxter operators on semigroup algebras of order two and three

In this paper, we determine all the Rota-Baxter operators of weight zero on semigroup algebras of order two and three with the help of computer algebra. We determine the matrices for these Rota-Baxter operators by directly solving the defining equations of the operators. We also produce a Mathematica procedure to predict and verify these solutions.     

Finite-dimensional solvable Lie algebras generated by normal operators are commutative

It is proved that any finite-dimensional solvable Lie algebra generated by normal operators is commutative, and some other related results are given.