On the linear Lindenbaum algebra of Basic Propositional Logic

We study the linear Lindenbaum algebra of Basic Propositional Calculus, called linear basic algebra. (© 2003 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)     

Varieties of Algebras without the Amalgamation Property

Let α be an ordinal and κ be a cardinal, both infinite, such that κ ≤ |α|. For τ ∈^αα, let sup(τ) = {i ∈ α: τ(i) ≠ i}. Let G _κ = {τ ∈^αα: |sup(τ)| < κ}. We consider variants of polyadic equality algebras by taking cylindrifications on Γ ? α, |Γ| < κ and substitutions restricted to G _κ. Such algebras are also enriched with generalized diagonal elements. We show that for any variety V containing the class of representable algebas and satisfying a finite schema of equations, V fails to have the amalgamation property. In particular, many varieties of Halmos’ quasi-polyadic equality algebras and Lucas’ extended cylindric algebras (including that of the representable algebras) fail to have the amalgamation property.     

Decision methods for linearly ordered Heyting algebras

The decision problem for positively quantified formulae in the theory of linearly ordered Heyting algebras is known, as a special case of work of Kreisel, to be solvable; a simple solution is here presented, inspired by related ideas in Gödel-Dummett logic.     

On the radical for some class of Banach algebras

Let A be a complex Banach algebra. It is well known that the second dual A** of A can be equipped with a multiplication that extends the original multiplication on A and makes A** a Banach algebra. We show that Rad(A) = ^⊥(A ^* · A) and Rad(A ^**) = (A ^* · A)^⊥ for some classes of Banach algebras A with scattered structure space. Some applications of these results are given.     

On the class equation for Hopf algebras

We give a simple proof of the Kac-Zhu class equation for semisimple Hopf algebras over an algebraically closed field of characteristic 0.

     

格蕴涵代数的LI-理想格及其素元刻画

运用格理论的原理和方法对格蕴涵代数L的LI-理想概念作进一步研究.首先,在L的全体LI-理想之集ф_(LI)(L)上定义了格运算■和■,蕴涵运算■以及伪补运算■,证明了(ф_(LI)(L),,■,■,■,{O},L)构成一个完备Heyting代数的结论.其次,利用运算固的性质给出了(ф_(LI)(L),,■,■,■,■,{O},L)成为Boolean代数的若干充要条件.最后,借助于L的素LI-理想之特性获得了格(ф_(LI)(L),,■,■,{O},L)中素元的若干等价刻画.     

扰动模糊命题逻辑的代数结构及其广义重言式性质

着眼于扰动模糊命题逻辑的代数结构,为研究二维扰动模糊命题逻辑最大子代数I2R及其广义重言式提供了一些代数理论基础,最后研究了子代数间广义重言式的关系.     

A survey of fuzzy implication algebras and their axiomatization

The theory of fuzzy implication algebras was proposed by Professor Wangming Wu in 1990. The present paper reviews the following two aspects of studies on FI-algebras: concepts, properties and some subclasses of FI-algebras; axiomatization of the class of FI-algebras and some of its important subclasses. The main results are summarized in the current paper, the relationships between FI-algebras and several classes of important fuzzy algebras are discussed, such as BL-algebras, MTL-algebras, and residuated lattices, and propositional calculus systems of several special classes of FI-algebras are shown.     

On the weak Freese-Nation property of complete Boolean algebras

The following results are proved:

(a) In a model obtained by adding ₂ Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □_μ and cof([μ]^₀,)=μ⁺ hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of is equivalent to the weak Freese-Nation property of any of or for uncountable κ. (d) Modulo the consistency of (_ω+1,_ω)(₁,₀), it is consistent with GCH that does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding _ω Cohen reals destroys the weak Freese-Nation property of .

These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.     

Confluent operator algebras and the closability property

Certain operator algebras A on a Hilbert space have the property that every densely defined linear transformation commuting with A is closable. Such algebras are said to have the closability property. They are important in the study of the transitive algebra problem. More precisely, if A is a two-transitive algebra with the closability property, then A is dense in the algebra of all bounded operators, in the weak operator topology. In this paper we focus on algebras generated by a completely nonunitary contraction, and produce several new classes of algebras with the closability property. We show that this property follows from a certain strict cyclicity property, and we give very detailed information on the class of completely nonunitary contractions satisfying this property, as well as a stronger property which we call confluence.     

Normalized integral table algebras generated by a faithful real element of degree 2 and having 4 linear elements

We completely classify the normalized integral table algebra (A, B) generated by a faithful real element of degree 2 and having four linear elements.     

The structure of Leavitt path algebras and the Invariant Basis Number property

In this paper, we provide the structure of the Leavitt path algebra of a finite graph via some step-by-step process of source eliminations, and restate Kanuni and Özaydin's nice criterion for Leavitt path algebras of finite graphs having Invariant Basis Number via matrix-theoretic language. Consequently, we give a matrix-theoretic criterion for the Leavitt path algebra of a finite graph having Invariant Basis Number in terms of a sequence of source eliminations. Using these results, we show certain classes of finite graphs for which Leavitt path algebras have Invariant Basis Number, as well as investigate the Invariant Basis Number property of Leavitt path algebras of certain Cayley graphs of finite groups.     

The class of infinite dimensional neat reducts of quasi‐polyadic algebras is not axiomatizable

SC, CA, QA and QEA denote the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasi‐polyadic algebras and quasi‐polyadic equality algebras, respectively. Let ω ≤ α < β and let K ∈ {SC,CA,QA,QEA}. We show that the class of α ‐dimensional neat reducts of algebras in K_β is not elementary. This solves a problem in [3]. Also our result generalizes results proved in [2] and [3]. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Complexity of some natural problems on the class of computable I-algebras

We study computable Boolean algebras with distinguished ideals (I-algebras for short). We prove that the isomorphism problem for computable I-algebras is Σ ₁ ¹ -complete and show that the computable isomorphism problem and the computable categoricity problem for computable I-algebras are Σ ₃ ⁰ -complete.     

Characterizations of isomorphisms and derivations of some algebras

Let ? be a zero-product preserving bijective bounded linear map from a unital algebra A onto a unital algebra B such that ?(1)=k. We show that if A is a CSL algebra on a Hilbert space or a J-lattice algebra on a Banach space then there exists an isomorphism ψ from A onto B such that ?=kψ. For a nest algebra A in a factor von Neumann algebra, we characterize the linear maps on A such that δ(x)y+xδ(y)=0 for all x,y∈A with xy=0.     

On the antipode of Hopf algebras with the dual Chevalley property

In this paper, we study the antipode of a finite-dimensional Hopf algebra H with the dual Chevalley property and obtain an annihilation polynomial for the antipode. This generalizes an old result given by Taft and Wilson in 1974. As consequences, we show that 1) the quasi-exponent of H is the same as the exponent of its coradical, that is, $\mathrm{qexp}(H)=\exp ?(H_0)$; 2) $\mathrm{qexp}(H?\mathbb{k}〈S^2〉)=\mathrm{qexp}(H)$.