Characterizations of regular ordered semigroups in terms of (α, β)-fuzzy generalized bi-ideals

In this paper, the concept of an (α, β)-fuzzy generalized bi-ideal in an ordered semigroup is introduced, which is a generalization of the concept of a fuzzy generalized bi-ideal in an ordered semigroup. Using this concept, some characterization theorems are provided. The upper/lower parts of an (∈, ∈ ∨ q)-fuzzy generalized bi-ideal are introduced and some characterizations of regular ordered semigroups are given. Also, we consider the concept of implication-based fuzzy generalized bi-ideals in an ordered semigroup. In particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.     

赋范格H蕴涵代数和模糊格H蕴涵代数

在范数的条件下扩充了格H蕴涵代数的概念,即赋范格H蕴涵代数,并讨论一些性质。然后将模糊集合论运用于赋范格H蕴涵代数,给出了模糊赋范格H蕴涵代数的定义,得到了一些基本性质。通过使用两个赋范格H蕴涵代数之间的映射定义了赋范格H蕴涵代数同态,且得到了一些性质。最后得到了在赋范格H蕴涵代数中的数列对于蕴涵距离是有界的结论。     

Approximations in <Emphasis Type="Italic">n</Emphasis>-ary algebraic systems

Algebraic systems have many applications in the theory of sequential machines, formal languages, computer arithmetics, design of fast adders and error-correcting codes. The theory of rough sets has emerged as another major mathematical approach for managing uncertainty that arises from inexact, noisy, or incomplete information. This paper is devoted to the discussion of the relationship between algebraic systems, rough sets and fuzzy rough set models. We shall restrict ourselves to algebraic systems with one n-ary operation and we investigate some properties of approximations of n-ary semigroups. We introduce the notion of rough system in an n-ary semigroup. Fuzzy sets, a generalization of classical sets, are considered as mathematical tools to model the vagueness present in rough systems.     

Approximations in a semigroup by using a neighbourhood system

Let S be a semigroup. We consider the relation γ and its transitive closure γ*. The relation γ is the smallest equivalence relation on S so that S/γ* is a commutative semigroup. Based on the relation γ, we define a neighbourhood system for each element of S, and we present a general framework of the study of approximations in semigroups. The connections between semigroups and operators are examined.     

乘积格蕴涵代数的子代数的MATLAB实现

借助于MATLAB,得到了两个有限链型的乘积格蕴涵代数子代数的具体形式,更直观地反映了格蕴涵子代数的结构特征,并且通过一个例子展示了程序的运行结果。研究了格蕴涵代数中各种滤子之间的关系,讨论了两个链型格蕴涵子代数乘积的滤子,得到了其只存在平凡滤子和素滤子的结论。     

On finite semigroups of matrices

Finite semigroups of n by n matrices over the naturals are characterized both by algebraic and combinatorial methods. Next we show that the cardinality of a finite semigroup S of n by n matrices over a field is bounded by a function depending only on n, the number of generators of S and the maximum cardinality of its subgroups. As a consequence, given n and k, there exist, up to isomorphism, only a finite number of finite semigroups of n by n matrices over the rationals, generated by at most k elements. Among other applications to Automaton Theory, we show that it is decidable whether the behavior of a given N – Σ automaton is bounded.     

一种泛逻辑代数系统

文[3]给出了理想状态(广义相关系数h=0.5,广义自相关系数k=0.5)下泛逻辑的形式演绎系统B,证明了此系统是可靠的。该文提出理想状态下(h=k=0.5)泛逻辑学对应的代数系统-UB代数,给出它的一系列性质。证明了UB代数是一个交换剩余半群;进一步证明了U B代数与M V代数、正规FI代数是等价的。     

Fuzzy radicals and prime fuzzy ideals of ordered semigroups

Let S be an ordered semigroup. A fuzzy subset of S is, by definition, an arbitrary mapping f: S → [0, 1], where [0, 1] is the usual interval of real numbers. Motivated by studying prime fuzzy ideals in rings, semigroups and ordered semigroups, and as a continuation of Kehayopulu and Tsingelis’s works in ordered semigroups in terms of fuzzy subsets, in this paper we introduce the notion of ordered fuzzy points of an ordered semigroup S, and give a characterization of prime fuzzy ideals of an ordered semigroup S. We also introduce the concepts of weakly prime fuzzy ideals, completely prime fuzzy ideals, completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of an ordered semigroup S, and establish the relationship between the five classes of ideals. Furthermore, we characterize weakly prime fuzzy ideals, completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of S by their level ideals. Finally, we introduce and investigate the fuzzy radicals of ordered semigroups by means of ordered fuzzy points, and prove that the fuzzy radical of every completely semiprime fuzzy ideal of an ordered semigroup S can be expressed as the intersection of all weakly completely prime fuzzy ideals containing it. As an application of the results of this paper, the corresponding results of semigroups (without order) are also obtained.     

A Fan-type result on k-ordered graphs

For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. In this paper, we show that if G is a ⌊3k/2⌋-connected graph of order n?100k, and d(u)+d(v)?n for any two vertices u and v with d(u,v)=2, then G is k-ordered hamiltonian. Our result implies the theorem of G. Chen et al. [Ars Combin. 70 (2004) 245-255] [1], which requires the degree sum condition for all pairs of non-adjacent vertices, not just those distance 2 apart.     

Linguistic truth-valued lattice-valued propositional logic system ?P(X) based on linguistic truth-valued lattice implication algebra

In the semantics of natural language, quantification may have received more attention than any other subject, and syllogistic reasoning is one of the main topics in many-valued logic studies on inference. Particularly, lattice-valued logic, a kind of important non-classical logic, can be applied to describe and treat incomparability by the incomparable elements in its truth-valued set. In this paper, we first focus on some properties of linguistic truth-valued lattice implication algebra. Secondly, we introduce some concepts of linguistic truth-valued lattice-valued propositional logic system ?P(X), whose truth-valued domain is a linguistic truth-valued lattice implication algebra. Then we investigate the semantic problem of ?P(X). Finally, we further probe into the syntax of linguistic truth-valued lattice-valued propositional logic system ?P(X), and prove the soundness theorem, deduction theorem and consistency theorem.     

Soft ideals and soft filters of soft ordered semigroups

It is shown that soft ordered semigroups over an ordered semigroup are actually soft ordered subsemigroups of (S,A). Soft idealistic semigroups over an ordered semigroup are soft ideals of soft ordered semigroup (S,A). The concept of soft filters of a soft ordered semigroup is introduced. It is shown that restricted complement of a soft filter is a soft prime ideal of (S,A).     

Notes on redefined fuzzy implicative filters of lattice implication algebras

The concept of -fuzzy implicative filters of lattice implication algebras is introduced and some of its related properties are investigated. In particular, the relationships among the ordinary fuzzy implicative filters, the (∈,∈∨q)-fuzzy implicative filters and the -fuzzy implicative filters of lattice implication algebras will be studied.     

The pseudo-linear semantics of interval-valued fuzzy logics

Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called interval-valued residuated lattices (IVRLs). Triangle algebras have been used to construct triangle logic (TL), a formal fuzzy logic that is sound and complete w.r.t. the class of IVRLs.In this paper, we prove that the so-called pseudo-prelinear triangle algebras are subdirect products of pseudo-linear triangle algebras. This can be compared with MTL-algebras (prelinear residuated lattices) being subdirect products of linear residuated lattices.As a consequence, we are able to prove the pseudo-chain completeness of pseudo-linear triangle logic (PTL), an axiomatic extension of TL introduced in this paper. This kind of completeness is the analogue of the chain completeness of monoidal T-norm based logic (MTL).This result also provides a better insight in the structure of triangle algebras; it enables us, amongst others, to prove properties of pseudo-prelinear triangle algebras more easily. It is known that there is a one-to-one correspondence between triangle algebras and couples (L,α), in which L is a residuated lattice and α an element in that residuated lattice. We give a schematic overview of some properties of pseudo-prelinear triangle algebras (and a number of others that can be imposed on a triangle algebra), and the according necessary and sufficient conditions on L and α.     

Regular ordered semigroups in terms of fuzzy subsets

Given a set S, a fuzzy subset of S (or a fuzzy set in S) is, by definition, an arbitrary mapping f : S → [0, 1] where [0, 1] is the usual interval of real numbers. If the set S bears some structure, one may distinguish some fuzzy subsets of S in terms of that additional structure. This important concept of a fuzzy set was first introduced by Zadeh. Fuzzy groups have been first considered by Rosenfeld, fuzzy semigroups by Kuroki. A theory of fuzzy sets on ordered groupoids and ordered semigroups can be developed. Some results on ordered groupoids-semigroups have been already given by the same authors in [N. Kehayopulu, M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum 65 (2002) 128-132; N. Kehayopulu, M. Tsingelis, The embedding of an ordered groupoid into a poe-groupoid in terms of fuzzy sets, Inform. Sci. 152 (2003) 231-236; N. Kehayopulu, M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Inform. Sci. 171 (2004) 13-28] where S has been endowed with the structure of an ordered semigroup and defined “fuzzy” analogous for several notions that have been proved to be useful in the theory of ordered semigroups. The characterization of regular rings in terms of right and left ideals is well known. The characterization of regular semigroups and regular ordered semigroups in terms of left and right ideals or in terms of left, right ideals and quasi-ideals is well known as well. The characterization of regular le-semigroups (that is lattice ordered semigroups having a greatest element) in terms of right ideal elements and left ideal elements or right, left and quasi-ideal elements is also known. In the present paper we first give the main theorem which characterizes the regular ordered semigroups by means of fuzzy right and fuzzy left ideals. Then we characterize the regular ordered semigroups in terms of fuzzy right, fuzzy left ideals and fuzzy quasi-ideals. The paper serves as an example to show that one can pass from the theory of ordered semigroups to the theory of “fuzzy” ordered semigroups. Some of our results are true for ordered groupoids in general.     

基于Clifford半群上共轭搜索问题的密钥建立协议

本文采用更广泛的半群作为平台，推广了Iris Anshel等提出的代数密钥建立协议模型。在定义了Clifford半群上的多重同时共轭搜索问题（MSCSP）后，给出了基于此问题的密钥建立协议。在理论上证明了若Clifford半群上的多重同时共轭搜索问题（MSCSP）是困难的，那么可以利用MSCSP来构造密钥建立协议。从而说明利用半群作为平台构建密钥建立协议是可能的。本文也提供了一种新的利用辫群的思路，即考虑利用辫群上的强半格（许多辨群按照一定规则形成的无交并）构成的Clifford半群来构建密码协议，以弥补单个辫群可能存在的安全缺陷。     

Revising the link between L-Chu correspondences and completely lattice L-ordered sets

Continuing our categorical study of L-fuzzy extensions of formal concept analysis, we provide a representation theorem for the category of L-Chu correspondences between L-formal contexts and prove that it is equivalent to the category of completely lattice L-ordered sets.     

Continuous approximations of product implication in MV-algebras with product

Recently, MV-algebras with product have been investigated from different points of view. In particular, in [EGM01], a variety resulting from the combination of MV-algebras and product algebras (see [H98]) has been introduced. The elements of this variety are called ŁΠ-algebras. Even though the language of ŁΠ-algebras is strong enough to describe the main properties of product and of ukasiewicz connectives on [0,1], the discontinuity of product implication introduces some problems in the applications, because a small error in the data may cause a relevant error in the output. In this paper we try to overcome this difficulty, substituting the product implication by a continuous approximation of it. We investigate the resulting algebras, called ŁΠ_q -algebras, and we show that these algebras constitute a quasivariety which is generated by the class of all ŁΠ _q -algebras whose lattice reduct is the unit interval [0,1] with the usual order.     

ON SEMANTICS OF L-VALUED FIRST-ORDER LOGIC Lvft

Abstract

Many-valued logic system always plays a crucial role in artificial intelligence. Many researchers have paid considerable attention to lattice-valued logic with truth values in a lattice. In this paper, based on lattice implication algebras introduced by Xu (Journal of Southwest Jiaolong University (in Chinese), Sum. No. 89(1), 20-27, 1993, and L-valued propositional logic _vft, established by Xu et al. (Information Sciences, 114, 20S-235, 1999a), the semantics of a L-type lattice-valued first-order logic L_vft, with truth values in lattice implication algebras were investigated. Some basic concepts about semantics of L_vftsuch as the language and the interpretation etc. were given and some semantic properties also were discussed. Finally, a concept of g-Skolem standard form was introduced, and it was shown that the unsatisfiability of a given lattice-valued formula was equivalent to that of its g-Skolem standard form. It will become a foundation to investigate the resolution principle based on first-order logic L_vft