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)     

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)     

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)     

Free‐decomposability in varieties of semi‐Heyting algebras

In this paper we prove that the free algebras in a subvariety $\mathcal V$ of the variety $\mathcal {SH}$ of semi‐Heyting algebras are directly decomposable if and only if $\mathcal V$ satisfies the Stone identity.     

On the free implicative semilattice extension of a Hilbert algebra

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.     

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 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)     

De Morgan Heyting algebras satisfying the identity xn(′*) ≈ x(n+1)(′*)

In this paper we investigate the sequence of subvarieties $ {\mathcal {SDH}_n} $of De Morgan Heyting algebras characterized by the identity x^n(′*) ≈ x^(n+1)(′*). We obtain necessary and sufficient conditions for a De Morgan Heyting algebra to be in $ {\mathcal {SDH}_1} $ by means of its space of prime filters, and we characterize subdirectly irreducible and simple algebras in $ {\mathcal {SDH}_1} $. We extend these results for finite algebras in the general case $ {\mathcal {SDH}_n} $. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the complexity of axiomatizations of the class of representable quasi‐polyadic equality algebras

Using games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEA _α of representable quasi‐polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka’s methods of splitting, to adapt the quasi‐polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEA _n for $2< n <\omegaUsing games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEA _α of representable quasi‐polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka’s methods of splitting, to adapt the quasi‐polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEA _n for $2< n <\omega$ and $l< n,$ $k < n$, k′ < ω are natural numbers, then Σ contains infinitely equations in which ? occurs, one of + or · occurs, a diagonal or a permutation with index l occurs, more than k cylindrifications and more than k′ variables occur. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

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)     

Powers of positive elements in C*‐algebras

In this paper, we show that Ogasawa’s theorem has a proof in Bishop style constructive mathematics (BISH). In 25 , we introduced the elementary constructive theory of C*‐algebras in BISH, but we did not discuss the powers of positive elements there. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Binary types in ℵ0‐categorical weakly o‐minimal theories

Orthogonality of all families of pairwise weakly orthogonal 1‐types for ?₀‐categorical weakly o‐minimal theories of finite convexity rank has been proved in 6 . Here we prove orthogonality of all such families for binary 1‐types in an arbitrary ?₀‐categorical weakly o‐minimal theory and give an extended criterion for binarity of ?₀‐categorical weakly o‐minimal theories (additionally in terms of binarity of 1‐types). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Low sets without subsets of higher many‐one degree

Given a reducibility ?_r, we say that an infinite set A is r‐introimmune if A is not r‐reducible to any of its subsets B with |A\B| = ∞. We consider the many‐one reducibility ?_m and we prove the existence of a low₁ m‐introimmune set in Π⁰₁ and the existence of a low₁ bi‐m‐introimmune set.     

On quasi‐varieties of multiple valued logic models

We extend the concept of quasi‐variety of first‐order models from classical logic to multiple valued logic (MVL) and study the relationship between quasi‐varieties and existence of initial models in MVL. We define a concept of ‘Horn sentence’ in MVL and based upon our study of quasi‐varieties of MVL models we derive the existence of initial models for MVL ‘Horn theories’. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Congruences and ideals on Boolean modules: a heterogeneous point of view

Definitions for heterogeneous congruences and heterogeneous ideals on a Boolean module $\mathcal {M}$ are given and the respective lattices $\mathrm{Cong}\mathcal {M}$ and $\mathrm{Ide}\mathcal {M}$ are presented. A characterization of the simple bijective Boolean modules is achieved differing from that given by Brink in a homogeneous approach. We construct the smallest and the greatest modular congruence having the same Boolean part. The same is established for modular ideals. The notions of kernel of a modular congruence and the congruence induced by a modular ideal are introduced to describe an isomorphism between $\mathrm{Cong}\mathcal {M}$ and $\mathrm{Ide}\mathcal {M}$. This isomorphism leads us to conclude that the class of the Boolean module is ideal determined.     

Rice and Rice‐Shapiro Theorems for transfinite correction grammars

Hay and, then, Johnson extended the classic Rice and Rice‐Shapiro Theorems for computably enumerable sets, to analogs for all the higher levels in the finite Ershov Hierarchy. The present paper extends their work (with some motivations presented) to analogs in the transfinite Ershov Hierarchy. Some of the transfinite cases are done for all transfinite notations in Kleene's important system of notations, $\mathcal {O}$. Other cases are done for all transfinite notations in a very natural, proper subsystem $\mathcal {O}_{\mathrm{Cantor}}$ of $\mathcal {O}$, where $\mathcal {O}_{\mathrm{Cantor}}$ has at least one notation for each constructive ordinal. In these latter cases it is open as to what happens for the entire set of transfinite notations in $(\mathcal {O} -\mathcal {O}_{\mathrm{Cantor}})$.     

A class of non‐graded left‐symmetric algebraic structures on the Witt algebra

We classify the compatible left‐symmetric algebraic structures on the Witt algebra satisfying certain non‐graded conditions. It is unexpected that they are Novikov algebras. Furthermore, as applications, we study the induced non‐graded modules of the Witt algebra and the induced Lie algebras by Novikov‐Poisson algebras’ approach and Balinskii‐Novikov's construction.     

Norm‐controlled inversion in smooth Banach algebras,II

We show that smoothness implies norm‐controlled inversion: the smoothness of an element a in a Banach algebra with a one‐parameter automorphism group is preserved under inversion, and the norm of the inverse is controlled by the smoothness of a and by spectral data. In our context smooth subalgebras are obtained with the classical constructions of approximation theory and resemble spaces of differentiable functions, Besov spaces or Bessel potential spaces. To treat ultra‐smoothness, we resort to Dales‐Davie algebras. Furthermore, based on Baskakov's work, we derive explicit norm control estimates for infinite matrices with polynomial off‐diagonal decay. This is a quantitative version of Jaffard's theorem.