**6. Conclusions**

Liu and Shi [11] introduced *M*-hazy groups by using the *M*-hazy binary operation. Mehmood et al. [12,13] extended this idea by defining *M*-hazy rings and obtained its induced fuzzifying convexities. By getting the motivations of these new proposed concepts through M-hazy operations, we proposed a new generalization of vector spaces over field based on *M*-hazy binary operation, which is denoted as *M*-hazy vector spaces over *M*-hazy field. In addition, by using the completely residuated lattice valed logic, some important properties of *M*-hazy field, *M*-hazy vector space, and *M*-hazy subspace are introduced. Based on these properties, it is shown that an *M*-hazy subspace of *M*-hazy vector space forms a convex structure. In addition, the linear transformation of *M*-hazy vector space is defined and proves its important results. The convexity of the *M* fuzzy set on the set *P* is the *M* fuzzy set on the power set with certain properties. Therefore, to a certain extent, each subset of *P* can be regarded as a convex set. Finally, considering the importance of this fact, a method is given that uses the *M*-hazy subspace of *M*-hazy vector space to induce the *M*-fuzzifying convex space.

The possible directions for the future work could be *M*-hazy modules, bases, dimensions of *M*-hazy vector spaces, *M*-hazy topological vector spaces, and other fuzzy algebra.

**Author Contributions:** Conceptualization, F.M.; formal analysis, F.M. and F.-G.S.; funding acquisition, F.-G.S.; investigation, F.M. and F.-G.S.; methodology, F.M. and F.-G.S.; project administration, F.M. and F.-G.S.; supervision, F.-G.S.; validation, F.M. and F.-G.S.; visualization, F.-G.S.; writing—original draft, F.M.; writing—review and editing, F.M. and F.-G.S. Both authors have read and agreed to the published version of the manuscript.

**Funding:** The project is funded by the National Natural Science Foundation of China (11871097).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank the editors and the anonymous reviewers for their fruitful comments and suggestions, which led to a number of improvements of the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
