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)     

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)     

Universal Classes of MV‐chains with Applications to Many‐valued Logics

In this paper we characterize, classify and axiomatize all universal classes of MV‐chains. Moreover, we accomplish analogous characterization, classification and axiomatization for congruence distributive quasivarieties of MV‐algebras. Finally, we apply those results to study some finitary extensions of the Łukasiewicz infinite valued propositional calculus.     

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)     

New types of hyper MV‐deductive systems in hyper MV‐algebras

The notions of a (weak) hyper MV‐deductive system, a (?, ?; ?)‐hyper MV‐deductive system, a (?, ?; ?)‐ hyper MV‐deductive system, a (?, ?; ?)‐hyper MV‐deductive system, a (?, ?; ?)‐hyper MV‐deductive system and a (∩, ∩; ∩)‐hyper MV‐deductive system are introduced, and then their relations are investigated (© 2010 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)     

σ‐short Boolean algebras

We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Vaught’s Conjecture and finitely valued MV algebras

We show that the complete first order theory of an MV algebra has $2^{\aleph _0}$ countable models unless the MV algebra is finitely valued. So, Vaught's Conjecture holds for all MV algebras except, possibly, for finitely valued ones. Additionally, we show that the complete theories of finitely valued MV algebras are $2^{\aleph _0}$ and that all ω‐categorical complete theories of MV algebras are finitely axiomatizable and decidable. As a final result we prove that the free algebra on countably many generators of any locally finite variety of MV algebras is ω‐categorical.     

On the Boolean algebras of definable sets in weakly o‐minimal theories

We consider the sets definable in the countable models of a weakly o‐minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic (hence T is p‐ω‐categorical), in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete (convex) subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within expansions of Boolean lattices, every weakly o‐minimal theory is p‐ω‐categorical. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Notes on local o‐minimality

We introduce and study some local versions of o‐minimality, requiring that every definable set decomposes as the union of finitely many isolated points and intervals in a suitable neighbourhood of every point. Motivating examples are the expansions of the ordered reals by sine, cosine and other periodic functions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

ω‐categorical weakly o‐minimal expansions of Boolean lattices

We study ω‐categorical weakly o‐minimal expansions of Boolean lattices. We show that a structure ?? = (A,≤, ?) expanding a Boolean lattice (A,≤) by a finite sequence I of ideals of A closed under the usual Heyting algebra operations is weakly o‐minimal if and only if it is ω‐categorical, and hence if and only if A/I has only finitely many atoms for every I ∈ ?. We propose other related examples of weakly o‐minimal ω‐categorical models in this framework, and we examine the internal structure of these models.     

Hierarchies in φ‐spaces and applications

We establish some results on the Borel and difference hierarchies in φ‐spaces. Such spaces are the topological counterpart of the algebraic directed‐complete partial orderings. E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non‐collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space Pω. We also sketch some older applications of these hierarchies and present a new application to the question of characterizing the ω‐ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bernstein sets and κ ‐coverings

In this paper we study a notion of a κ ‐covering set in connection with Bernstein sets and other types of non‐measurability. Our results correspond to those obtained by Muthuvel in [7] and Nowik in [8]. We consider also other types of coverings (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global minimum and orthogonality in Cp ‐classes

In this paper we use recent results [14] to establish various characterizations of the global minimum of the map F_ψ : U → ?⁺ defined by F_ψ (X) = ‖ψ (X)‖_p (1 < p < ∞) where ψ: U → C_p is a map defined by ψ (X) = S +? (X), with ?: B (H) → B (H) a linear map and S ∈ C_p , and U = {X ∈ B (H): ? (X) ∈ C_p }. Further, we apply these results to characterize the operators which are orthogonal to the range of elementary operators. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pseudo asymptotically periodic solutions for Volterra integro‐differential equations

In this paper, we propose a new class of functions called pseudo ‐asymptotically ω‐periodic function in the Stepanov sense and explore its properties in Banach spaces including composition results. Furthermore, the existence and uniqueness of the pseudo ‐asymptotically ω‐periodic mild solutions to Volterra integro‐differential equations is investigated. Applications to integral equations arising in the study of heat conduction in materials with memory are shown. Copyright © 2014 John Wiley & Sons, Ltd.     

On some sets of dictionaries whose ω ‐powers have a given

A dictionary is a set of finite words over some finite alphabet X. The ω ‐power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [10] the complexity of the set of dictionaries whose associated ω ‐powers have a given complexity. In particular, he considered the sets ??( Σ ⁰k) (respectively, ??( Π ⁰_k), ??( Δ ¹₁)) of dictionaries V ? 2* whose ω ‐powers are Σ ⁰_k‐sets (respectively, Π ⁰_k‐sets, Borel sets). In this paper we first establish a new relation between the sets ??( Σ ⁰₂) and ??( Δ ¹₁), showing that the set ??( Δ ¹₁) is “more complex” than the set ??( Σ ⁰₂). As an application we improve the lower bound on the complexity of ??( Δ ¹₁) given by Lecomte, showing that ??( Δ ¹₁) is in Σ ¹ ₂(2^2*)\ Π ⁰₂. Then we prove that, for every integer k ≥ 2 (respectively, k ≥ 3), the set of dictionaries ??( Π ⁰_k+1) (respectively, ??( Σ ⁰_{k +1})) is “more complex” than the set of dictionaries ??( Π ⁰_k) (respectively, ??( Σ ⁰_k)) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On effective σ‐boundedness and σ‐compactness

We prove several dichotomy theorems which extend some known results on σ‐bounded and σ‐compact pointsets. In particular we show that, given a finite number of $\Delta ^{1}_{1}$ equivalence relations $\mathrel {\mathsf {F}}_1,\dots ,\mathrel {\mathsf {F}}_n$, any $\Sigma ^{1}_{1}$ set A of the Baire space either is covered by compact $\Delta ^{1}_{1}$ sets and lightface $\Delta ^{1}_{1}$ equivalence classes of the relations $\mathrel {\mathsf {F}}_i$, or A contains a superperfect subset which is pairwise $\mathrel {\mathsf {F}}_i$‐inequivalent for all i = 1, …, n. Further generalizations to $\Sigma ^{1}_{2}$ sets A are obtained.     

On (∈, ∈ ∨ q)‐fuzzy filters of R0‐algebras

In this paper, we introduce the notions of (∈, ∈ ∨ q)‐fuzzy filters and (∈, ∈ ∨ q)‐fuzzy Boolean (implicative) filters in R₀‐algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that a fuzzy set in R₀‐algebras is an (∈, ∈ ∨ q)‐fuzzy Boolean filter if and only if it is an (∈, ∈ ∨ q)‐fuzzy implicative filter. Finally, we consider the concepts of implication‐based fuzzy Boolean (implicative) filters of R₀‐algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the kernel space for an L0‐linear function

A random normed module is a random generalization of an ordinary normed space, and it is the randomization that makes a random normed module possess rich stratification structures. On the basis of these stratification structures, this paper shows that either the kernel space N(f) for an L⁰‐linear function f from a random normed module S to the algebra is a closed submodule or N(f) on some specifical stratification is a dense proper submodule of S, which generalizes the classical case. In the meantime, a characterization for the kernel space N(f) to be closed is also given.