### BIT Makes Achievement in Theoretical Study of Fuzzy Convex Space

**Beijing Institute of Technology, Dec 24 ^{th}, 2019: **Recently, Pang Bin and Shi Fugui, working as Associate Researcher and Professor at School of Mathematics and Statistics, Beijing Institute of Technology(BIT), published a study entitled “Fuzzy counterparts of hull operators and interval operators in the framework of L-convex spaces” in the international top academic journal

*Fuzzy Sets and Systems*. This paper studies the convex hull operators and interval operators in the framework of L-convex spaces. It is proved that the L-convex space category and the L-convex hull space category are isomorphic, as well as the strong L-convex space category and the L-ordered convex hull space category are isomorphic. At the same time, the proof of a Galois connection between the L-interval space category and the L-convex space category is given in the paper. As a reflection subcategory, the L-convex space category of arity2 can be embedded into the L-interval space category.

The concept of convex sets is widely utilized in various areas of study in mathematics. Corresponding convex sets have different forms under the different structure framework of mathematics. In order to study the similarities between these convex sets in several forms, the concept of convex structures is put into consideration. Convex structure is an axiomatic spatial structure obtained by abstracting the basic properties of different types of convex sets. The concept of fuzzy convex structure was first introduced in 1994 with the development of theoretical fuzzy mathematics. However, this theory has not widely paid attention to for a long time. The two fundamental tools for research in fuzzy convex space were not effectively solved, which resulted in the limited attention. Convex hull operators and interval operators are two basic concepts in the theory of convex structure. As an equivalent way of expression in terms of convex structure, the convex hull operator is used in researches of the separation properties, convex invariance properties of convex structures, and to establish the relationship between convex structures and other spatial structures. Interval operators are mainly used to describe the geometrical properties of convex structure. While in the framework of fuzzy convex structure, there is no acceptable definition of convex hull operators and interval operators, which caused few results of research in this theory without continuous development.

In 2019, Associate Researcher Pang Bin and Professor Shi Fugui introduced the concept of L-convex hull operators under the lattice-valued circumstance of complete lattices, defined L-order convex hull operators under the lattice-valued circumstance of complete residual lattices and gave the definition of L-interval operator under the lattice-valued circumstance of complete allocated lattices. At the same time, they systematically established the category relationship between these definitions and L-convex structure by utilizing the logic of value lattice. The related paper "Fuzzy counterparts of hull operators and interval operators in the framework of L-convex spaces" was published in the top journal *Fuzzy Sets and Systems*. This work successfully gave two most fundamental tools for studying L-convex structure theory, that is, L-convex hull operator and L-interval operator. And this study lays a solid theoretical foundation for further research on L-convex structure theory. The paper has attracted worldwide attention from scholars once it published, and has been selected into the ESI highly cited paper.

This work of research is completed by Associate Researcher Pang Bin and Professor Shi Fugui. Associate Researcher Pang Bin is the first author of the paper. This work is supported by National Natural Science Foundation of China.

**Link of the paper: **https://doi.org/10.1016/j.fss.2018.05.012

**Attachment: Brief introduction of the researcher and the team**

The fuzzy mathematics research team of School of Mathematics and Statistics, BIT has long been engaged in the research of fuzzy set theory, fuzzy algebra, fuzzy topology, fuzzy matroids, and fuzzy convex spaces in fuzzy mathematics. Professor Shi Fugui, the team leader, is a second-level professor, a director of the Chinese Operations Research Society, a vice chairman of the Beijing Operations Research Society, an executive director of the Beijing Mathematical Society and a vice chairman of the Fuzzy Mathematics and Fuzzy Systems Committee of the Chinese Systems Engineering Society. He also serves on the editorial board of the SCI journal *Iranian Journal of Fuzzy Systems*. The team has opened up several branches of fuzzy mathematics research. The fuzzy convex space theory is currently developing into a research hotspot in the field of fuzzy mathematics. Associate Researcher Pang Bin, doctoral supervisor, is a member of the team of fuzzy mathematics theory and application of School of Mathematics and Statistics, BIT. Ph.D. graduated from BIT. His main fields of research are fuzzy topology, fuzzy convex structure and rough set theory. As the first author, he has published more than 10 SCI papers in top journals in fuzzy mathematics such as *Fuzzy Sets and Systems*, *IEEE Transactions on Fuzzy Systems*, and *Information Sciences*.

**News Source: **School of Mathematics and Statistics

**Editor:** News Agency of BIT

**Translator: **Liu Luchen, News Agency of BIT