*Article M***-Hazy Vector Spaces over** *M***-Hazy Field**

**Faisal Mehmood and Fu-Gui Shi \***

> Beijing Key Laboratory on MCAACI, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 102488, China; 3820170030@bit.edu.cn or faisal.mehmood007@gmail.com **\*** Correspondence: fuguishi@bit.edu.cn

**Abstract:** The generalization of binary operation in the classical algebra to fuzzy binary operation is an important development in the field of fuzzy algebra. The paper proposes a new generalization of vector spaces over field, which is called *M*-hazy vector spaces over *M*-hazy field. Some fundamental properties of *M*-hazy field, *M*-hazy vector spaces, and *M*-hazy subspaces are studied, and some important results are also proved. Furthermore, the linear transformation of *M*-hazy vector spaces is studied and their important results are also proved. Finally, it is shown that *M*-fuzzifying convex spaces are induced by an *M*-hazy subspace of *M*-hazy vector space.

**Keywords:** *M*-hazy group; *M*-hazy ring; *M*-hazy field; *M*-hazy vector space; *M*-hazy subspace; *M*-fuzzifying convex space
