R_0代数的滤子理论

在R0代数中引入了正蕴涵滤子、奇异滤子、MV滤子的概念,讨论了这些滤子的性质及关系.得到了:在R0代数上,蕴涵滤子、正蕴涵滤子、布尔滤子是等价的;奇异滤子与MV滤子是等价的;正蕴涵滤子是奇异滤子,但反之不真.     

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 (∈, ∈ ∨ 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 fuzzy fantastic filters of lattice implication algebras

Fuzzification of a fantastic filter in a lattice implication algebra is considered. Relations among a fuzzy filter, a fuzzy fantastic filter, and fuzzy positive implicative filter are stated. Conditions for a fuzzy filter to be a fuzzy fantastic filter are given. Using the notion of level set, a characterization of a fuzzy fantastic filter is considered. Extension property for fuzzy fantastic filters is established. The notion of normal/maximal fuzzy fantastic filters and complete fuzzy fantastic filters is introduced, and some related properties are investigated.     

On Special Implicative Filters

In her well-known book, Rasiowa states without proof that in implicative algebras there is a one-to-one correspondence between kernels of epimorphisms and the so-called special implicative filters, and that in the logic whose algebraic counterpart is the class of implicative algebras the deductive filters coincide with the special implicative filters. We show that neither claim is true, and how to repair the situation by redefining some of the notions involved. We answer other questions concerning special implicative filters, taking the theory of algebraizable logics of Blok and Pigozzi as a framework to approach the question in a systematic way.     

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)     

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.     

Some kinds of falling fuzzy filters of lattice implication algebras

In this paper, the concepts of falling fuzzy(implicative, associative) filters of lattice implication algebras based on the theory of falling shadows and fuzzy sets are presented at first. And then the relations between fuzzy(implicative, associative) filters and falling fuzzy(implicative, associative) filters are provided. In particular, we put forward an open question on a kind of falling fuzzy filters of lattice implication algebras. Finally, we apply falling fuzzy inference relations to lattice implication algebras and obtain some related results.     

Spectral-like duality for distributive Hilbert algebras with infimum

Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and \({\wedge}\)-semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.     

On convergence theory in fuzzy topological spaces and its applications

In this paper we introduce and study new concepts of convergence and adherent points for fuzzy filters and fuzzy nets in the light of the Q-relation and the Q-neighborhood of fuzzy points due to Pu and Liu [28]. As applications of these concepts we give several new characterizations of the closure of fuzzy sets, fuzzy Hausdorff spaces, fuzzy continuous mappings and strong Q-compactness. We show that there is a relation between the convergence of fuzzy filters and the convergence of fuzzy nets similar to the one which exists between the convergence of filters and the convergence of nets in topological spaces.     

剩余格中的滤子

This paper is devoted to the discussion of filters in residuated lattices. Some equivalent conditions about filter were given. The structure of generated filter was established. The concept of implicative filter in residuated lattice was proposed with its basic properties being discussed.     

Interval Valued (∈,∈ ∨q)-Fuzzy Filters of MTL-Algebras

The concepts of interval valued (∈,∈ ∨q)-fuzzy Boolean, MV - and G- filters in a MTL-algebra are introduced. The properties of these generalized fuzzy filters are studied and in particular, the relationships between these fuzzy filters in an MTL-algebra are investigated.     

Implicative Ideals and Fuzzy Implicative Ideals of a Distributive Implication Groupoid

We introduce the notions of implicative ideals and fuzzy implicative ideals of a distributive implication groupoid. Some properties of these ideals will be investigated. In particular, the necessary and sufficient conditions for an ideal (fuzzy ideal) to be an implicative ideal (fuzzy implicative ideal) is given. By using the concept of level sets, we will characterize the fuzzy implicative ideals of a distributive implication groupoid. Finally, an extension property for fuzzy implicative ideals is given.     

N(2,2,0)代数的(∈,∈∨q_(λ,μ))-模糊关联理想

从以下几个方面对N(2,2,0)代数(∈,∈∨q_((λ,μ)))-模糊关联理想进行了详细的研究.首先,给出N(2,2,0)代数广义模糊关联理想概念和点态化(∈,∈∨q_((λ,μ)))-模糊关联理想概念,研究了两者之间的等价关系;其次,给出了(∈,∈∨q_((λ,μ)))-模糊关联理想若干等价刻画;再次,讨论了(∈,∈∨q_((λ,μ)))-模糊关联理想与(∈,∈∨q_((λ,μ)))-模糊理想之间的关系;最后,获得了(∈,∈∨q_((λ,μ)))-模糊关联理想交与并的相关性质.     

Implicative hyper <Emphasis Type="Italic">K</Emphasis>-algebras

In this note we first define the notions of (weak, strong) implicative hyper K-algebras. Then we show by examples that these notions are different. After that we state and prove some theorems which determine the relationship between these notions and (weak) hyper K-ideals. Also we obtain some relations between these notions and (weak) implicative hyper K-ideals. Finally, we study the implicative hyper K-algebras of order 3, in particular we obtain a relationship between the positive implicative hyper K-algebras and (weak, strong) implicative hyper K-algebras under a simple condition.     

Homomorphisms of implicative semigroups

Implicative semigroups and Brouwerian semigroups were studied by W. C. Nemitz and T. S. Blyth, respectively. In this paper, following the ideas of Nemitz and Blyth, we introduce the notion of negatively partially ordered implicative semigroups and studied the homomorphisms between these semigroups. Some results of Nemitz on implicative semilattices are generalized and amplified to implicative semigroups. Research is partially supported by CUHK small Project grant 220.600.080.     

伪补MS-代数的滤子同余关系

在伪补MS-代数上引入余核滤子和完全滤子的概念,研究伪补MS-代数的余核滤子和完全滤子的性质,获得了余核滤子和完全滤子生成的同余关系的表达式,证明了具有余核滤子的最小同余关系有同余一致性.     

A Note on Hilbert Algebras and Their Related Generalized Esakia Spaces

Generalized Esakia spaces are the topological duals of bounded implicative semilattices in the duality studied by G. Bezhanishvili and R. Jansana. We study the relation between a Hilbert algebra and the generalized Esakia space dual to its free implicative semilattice extension. To establish the relation we introduce a category whose objects are a generalized Esakia space together with a family of clopen up-sets that constitutes a subalgebra of the implication fragment of the Heyting algebra of the up-sets of the generalized Esakia space.     

Girard Quantale中的模糊关联滤子

在Girard Quantale中引入了模糊滤子和模糊关联滤子的概念,研究了Girard Quantale中模糊滤子和模糊关联滤子的一些性质,给出了Girard Quantale中模糊关联滤子的若干等价刻画,探讨了模糊关联滤子与模糊布尔滤子之间的关系.     

正定关联BCI—代数

本文是作者［１］和［２］的继续，引入了正定关联ＢＣＩ－代数的概念，并证明了：正定关联ＧＢＣＫ－代数类和Ｐ－半单ＢＣＩ－代数类是正定关联ＢＣＩ－代数类的真子类。