Xianzhong Zhao

Bio: Xianzhong Zhao is an academic researcher from Northwest University (China). The author has contributed to research in topics: Semiring & Mathematics. The author has an hindex of 3, co-authored 3 publications receiving 557 citations.

Soft semirings

Abstract: Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we initiate the study of soft semirings by using the soft set theory. The notions of soft semirings, soft subsemirings, soft ideals, idealistic soft semirings and soft semiring homomorphisms are introduced, and several related properties are investigated.

*-μ-Semirings and *-μ-lamda;-semirings

Abstract: We introduce and study *-µ-semirings and *-λ-semirings which generalize inductive *-semirings and weak inductive *-semirings, respectively. Also, we discuss the semiring of formal power series with coefficients in such a semiring and prove that the semiring of formal power series with coefficients in a weak inductive *-semiring [µ-semiring, λ-semiring, *-λ-semiring] is a weak inductive *-semirng [µ-semiring, λ-semiring, *-λ-semiring, respectively]. This gives a positive answer to one of Esik and Kuich's open problems.

∗--semirings and ∗--semirings

Abstract: We introduce and study ∗--semirings and ∗--semirings which generalize inductive ∗-semirings and weak inductive ∗-semirings, respectively. Also, we discuss the semiring of formal power series with coefficients in such a semiring and prove that the semiring of formal power series with coefficients in a weak inductive ∗-semiring [-semiring, -semiring, ∗--semiring] is a weak inductive ∗-semirng [-semiring, -semiring, ∗--semiring, respectively]. This gives a positive answer to one of Esik and Kuich’s open problems. © 2005 Elsevier B.V. All rights reserved.

Flat extensions of groups and limit varieties of additively idempotent semirings

Abstract: The present paper is devoted to the study of limit varieties of additively idempotent semirings. A limit variety is a nonfinitely based variety whose proper subvarieties are all finitely based. We present concrete constructions for one infinite family of limit additively idempotent semiring varieties, and one further ad hoc example. Each of these examples can be generated by a finite flat semiring, with the infinite family arising by a way of a complete characterisation of limit varieties that can be generated by the flat extension of a finite group. We also demonstrate the existence of other examples of limit varieties of additively idempotent semirings, including one further continuum-sized family, each with no finite generator, and two further ad hoc examples. While an explicit description of these latter examples is not given, one of the examples is proved to contain only trivial flat semirings.

On the groups associated with a tropical n × n matrix

Abstract: In this paper, the generalized centralizer group Un(A) of a tropical n×n matrix A and the centralizer group Pn(E) of a tropical idempotent normal matrix E are introduced and studied. It is proved that Un(A) is a product of two specific normal subgroups. And a structural description of Pn(E) is given when E is not strongly regular. It is also made some observations on E when Pn(E) is isomorphic to a 2-closed transitive permutation group on {1,2,…,n}.

On some new operations in soft set theory

Abstract: Molodtsov introduced the theory of soft sets, which can be seen as a new mathematical approach to vagueness. In this paper, we first point out that several assertions (Proposition 2.3 (iv)-(vi), Proposition 2.4 and Proposition 2.6 (iii), (iv)) in a previous paper by Maji et al. [P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555-562] are not true in general, by counterexamples. Furthermore, based on the analysis of several operations on soft sets introduced in the same paper, we give some new notions such as the restricted intersection, the restricted union, the restricted difference and the extended intersection of two soft sets. Moreover, we improve the notion of complement of a soft set, and prove that certain De Morgan's laws hold in soft set theory with respect to these new definitions.

On soft topological spaces

Abstract: In the present paper we introduce soft topological spaces which are defined over an initial universe with a fixed set of parameters. The notions of soft open sets, soft closed sets, soft closure, soft interior points, soft neighborhood of a point and soft separation axioms are introduced and their basic properties are investigated. It is shown that a soft topological space gives a parametrized family of topological spaces. Furthermore, with the help of an example it is established that the converse does not hold. The soft subspaces of a soft topological space are defined and inherent concepts as well as the characterization of soft open and soft closed sets in soft subspaces are investigated. Finally, soft T"i-spaces and notions of soft normal and soft regular spaces are discussed in detail. A sufficient condition for a soft topological space to be a soft T"1-space is also presented.

Soft set theory and uni–int decision making

Abstract: We firstly redefine the operations of Molodtsov’s soft sets to make them more functional for improving several new results. We also define products of soft sets and uni–int decision function. By using these new definitions we then construct an uni–int decision making method which selects a set of optimum elements from the alternatives. We finally present an example which shows that the method can be successfully applied to many problems that contain uncertainties.

Soft sets combined with fuzzy sets and rough sets: a tentative approach

Abstract: Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft---rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.

Soft sets and soft rough sets

Abstract: In this study, we establish an interesting connection between two mathematical approaches to vagueness: rough sets and soft sets. Soft set theory is utilized, for the first time, to generalize Pawlak's rough set model. Based on the novel granulation structures called soft approximation spaces, soft rough approximations and soft rough sets are introduced. Basic properties of soft rough approximations are presented and supported by some illustrative examples. We also define new types of soft sets such as full soft sets, intersection complete soft sets and partition soft sets. The notion of soft rough equal relations is proposed and related properties are examined. We also show that Pawlak's rough set model can be viewed as a special case of the soft rough sets, and these two notions will coincide provided that the underlying soft set in the soft approximation space is a partition soft set. Moreover, an example containing a comparative analysis between rough sets and soft rough sets is given.