模糊蕴涵代数的结构特征

本文研究了模糊蕴涵代数的一些性质,给出了模糊蕴涵代数成为Heyting代数的一个条件,得到对模糊蕴涵代数的结构特征刻画,并给出了一个(2,0)型代数(X,→,0)成为模糊蕴涵代数的充分必要条件.     

MTL代数及由其MP滤子型商代数的局部有限性

首先,将局部有限方法引入到MTL代数中,提出了局部有限MTL代数的概念,给出了局部有限MTL代数的一些基本性质,证明了局部有限MTL代数的线性性质;其次,讨论了MTL代数的MP滤子的相关性质;最后,证明了由MP滤子诱导的商贷数M/F是局部有限的非退化MTL代数的充分必要条件是MP滤子是极大MP滤子;证明了每一个MTL代数可以嵌入到一族局部有限的MTL代数的直积MTL代数中。     

HEYTING代数与FUZZY蕴涵代数

Heyting代数是作为直觉主义命题逻辑的代数模型而引进的Fuzzy蕴涵代数是 [0 ,1]值逻辑的蕴函联结词的一种代数抽象 .本文给出Heyting代数的若干基本性质 ,并证明了Heyting代数是Fuzzy蕴涵代数 ,也是Heyting型Fuzzy蕴涵代数。     

关于Z-蕴涵代数

基于N-半单代数和格蕴涵代数、FI-代数, Wajsberg-代数、BCK-代数、BCI-代数、BCC-代数及MV- 代数等的关系[9],本文中,我们引入了Z-蕴涵代数的概念, 并讨论了它们的某些性质.     

模糊蕴涵格理论

模糊蕴涵代数,在文献中简称为FI代数,最初由吴望名先生于1990年提出,至今已经有许多研究成果.文中综述有关FI代数的概念,性质等主要研究工作,同时给出这类代数的一些新的性质.重点强调构成格结构的FI代数,称之为模糊蕴涵格,简称为FI格.这类代数结构与模糊逻辑中几个重要的代数系统具有紧密的联系,文中将揭示这些联系,一些重要的模糊逻辑代数系统都是FI格类的子类.另外,所有正则FI格构成代数簇,即等式代数类.这个代数簇将在模糊逻辑与近似推理中发挥重要的作用.     

分配的Fuzzy蕴涵代数

给出了分配的Fuzzy蕴涵代数的定义并探讨了其有关性质，接着本文证明了分配的Fuzzy蕴涵代数与Boole代数、正则的HFI代数是相互等价的，从而得到Boole代数的两个等价形式，并且证明了分配的Fuzzy蕴涵代数是BL代数，最后得到了FI代数成为Boole代数的几个充要条件。     

HEYTING代数与FUZZY蕴函代数

Heyting代数是作为直觉主义命题逻辑的代数模型而引起的，Fuzzy蕴涵代数是[0,1]值逻辑的蕴涵联结词的一种代数抽象。本文给出Heyting代数的若干基本性质，并证明了Heyting代数是Fuzzy蕴涵代数，也是Heyting型Fuzzy蕴涵代数。     

DR0代数:由De Morgan代数导出的正则剩余格

首先讨论了De Morgan代数与剩余格的关系，并引入强De Morgan代数的概念，讨论了它的基本性质．随后，将著名的R0蕴涵拓广到De Morgan代数上，称为广义R0蕴涵；证明了添加广义凰蕴涵和相应 算子后的De Morgan代数L成为剩余格的充要条件是L为强De Morgan代数，并由此引入D‰代数的概念．接着，研究了DR0代数与‰代数的关系，证明了以下结论：Boole代数是DR0代数；全序DR0代数和全序R0代数等价；DR0代数是R0代数当且仅当它满足预线性条件；无中点的DR0代数是BL代数当且仅当它是Boole代数．最后，举例说明了非D兄D代数的RD代数、以及非R0代数的DR0代数都是存在的．     

关于PFI-代数与剩余格

本文提出了一种强FI代数-PFI代数,并且深入研究了它的性质,借此进一步揭示了FI-代数和剩余格之间更加密切的联系,进而以FI-代数为基本框架建立了R0-代数、正则剩余格等逻辑系统的结构特征(包括对隅结构)及其相互关系．这种以FI-代数为基础来统一处理剩余格和R0-代数的方法,同样适合于格蕴涵代数和MV代数等代数结构,而且从中更能清楚地看出它们之间的密切联系,也将有助于对相应形式逻辑系统与模糊推理的研究．     

MV代数定义的蕴涵简化形式

通过对MV代数和Lukasiewicz命题演算系统的研究,我们对MV代数的定义进行了简化,并讨论了MV代数和其它代数之间的关系。主要结果是:(1)从蕴涵角度出发,给出了MV代数的两种简化定义;(2)提出了弱格蕴涵代数的概念,并证明了它与BR0代数等价;(3)证明了弱格蕴涵代数是正则Fuzzy蕴涵代数。     

Generalized valuations on MTL-algebras

In this paper, we study some kinds of generalized valuations on MTL-algebras, discuss the relationship between the cokernel of generalized valuations and types of filters on MTL-algebras. Then, we give some equivalent characterizations of positive implicative generalized valuations on MTL-algebras. Finally, we characterize the structure theory of quotient MTL algebras based on the congruence relation, which is constructed by generalized valuations. The results of this paper not only generalize related theories of generalized valuations, but also enrich the algebraic conclusion of probability measure, on algebras of triangular norm based fuzzy logic.     

On n ‐contractive fuzzy logics

It is well known that MTL satisfies the finite embeddability property. Thus MTL is complete w. r. t. the class of all finite MTL‐chains. In order to reach a deeper understanding of the structure of this class, we consider the extensions of MTL by adding the generalized contraction since each finite MTL‐chain satisfies a form of this generalized contraction. Simultaneously, we also consider extensions of MTL by the generalized excluded middle laws introduced in [9] and the axiom of weak cancellation defined in [31]. The algebraic counterpart of these logics is studied characterizing the subdirectly irreducible, the semisimple, and the simple algebras. Finally, some important algebraic and logical properties of the considered logics are discussed: local finiteness, finite embeddability property, finite model property, decidability, and standard completeness. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Central simple Poisson algebras

Poisson algebras are fundamental algebraic structures in physics and sym-plectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero. The Lie algebra structures of these Poisson algebras are in general not finitely-graded.     

On intermediate inquisitive and dependence logics: An algebraic study

This article provides an algebraic study of intermediate inquisitive and dependence logics. While these logics are usually investigated using team semantics, here we introduce an alternative algebraic semantics and we prove it is complete for all intermediate inquisitive and dependence logics. To this end, we define inquisitive and dependence algebras and we investigate their model-theoretic properties. We then focus on finite, core-generated, well-connected inquisitive and dependence algebras: we show they witness the validity of formulas true in inquisitive algebras, and of formulas true in well-connected dependence algebras. Finally, we obtain representation theorems for finite, core-generated, well-connected, inquisitive and dependence algebras and we prove some results connecting team and algebraic semantics.     

Forcing operators on MTL‐algebras

We study the forcing operators on MTL‐algebras, an algebraic notion inspired by the Kripke semantics of the monoidal t ‐norm based logic (MTL). At logical level, they provide the notion of the forcing value of an MTL‐formula. We characterize the forcing operators in terms of some MTL‐algebras morphisms. From this result we derive the equality of the forcing value and the truth value of an MTL‐formula (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kleene代数及相关半模结构

Kleene代数在理论计算机科学中具有基础而特殊的重要性,Kleene模、布尔模和动态代数等与Kleene代数密切相关的半模结构在程序的语义逻辑及推理中发挥着十分重要的作用.将半环和半模等代数系统作为基本构架,研究了理论计算机科学中的Kleene代数、Kleene模和归纳~*-半环等重要概念,并将这些对象统一为序~*-半环上称为归纳半模的代数结构.进一步,提出并讨论了弱归纳半模、伪归纳半模以及伪弱归纳半模等相关概念.     

A Structure Theorem for Free Temporal Algebras

In this paper an algebraic version for temporal algebras of the logical filtrations for modal and temporal logics is analysed. A structure theorem for free temporal algebras and also some results with regard to the variety of temporal algebras are obtained.     

EQ-algebras

We introduce a new class of algebras called EQ-algebras. An EQ-algebra has three basic binary operations (meet, multiplication and a fuzzy equality) and a top element. These algebras are intended to become algebras of truth values for a higher-order fuzzy logic (a fuzzy type theory, FTT). The motivation stems from the fact that until now, the truth values in FTT were assumed to form either an IMTL-, BL-, or MV-algebra, all of them being special kinds of residuated lattices in which the basic operations are the monoidal operation (multiplication) and its residuum. The latter is a natural interpretation of implication in fuzzy logic; the equivalence is then interpreted by the biresiduum, a derived operation. The basic connective in FTT, however, is a fuzzy equality and, therefore, it is not natural to interpret it by a derived operation. This defect is expected to be removed by the class of EQ-algebras introduced and studied in this paper. From the algebraic point of view, the class of EQ-algebras generalizes, in a certain sense, the class of residuated lattices and so, they may become an interesting class of algebraic structures as such.     

On triangular norm based axiomatic extensions of the weak nilpotent minimum logic

In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic (WNM) and its t‐norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM‐algebras (?????) and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM‐chains, in particular varieties generated by standard WNM‐chains, or equivalently t‐norm based axiomatic extensions of WNM, and we study their standard completeness properties. We also characterize the generic WNM‐chains, i. e. those that generate the variety ?????, and we give finite axiomatizations for some t‐norm based extensions of WNM. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lie algebras with an algebraic adjoint representation revisited

A well-known theorem due to E. Zelmanov proves that PI-Lie algebras with an algebraic adjoint representation over a field of characteristic zero are locally finite-dimensional. In particular, a Lie algebra (over a field of characteristic zero) whose adjoint representation is algebraic of bounded degree is locally finite-dimensional. In this paper it is proved that a prime nondegenerate PI-Lie algebra with an algebraic adjoint representation over a field of characteristic zero is simple and finite-dimensional over its centroid, which is an algebraic field extension of the base field. We also give a new and shorter proof of the local finiteness of Lie algebras with an algebraic adjoint representation of bounded degree.