**1. Introduction**

In 1971, Rosenfeld [1] published an innovative paper on fuzzy subgroups. This article introduced the new field of abstract algebra and the new field of fuzzy mathematics. Many scientists and researchers worked in this field and obtained fruitful research. Liu [2,3] gave an important generalization in the field of fuzzy algebra by introducing fuzzy subrings of a ring and fuzzy ideals. Demirci [4] firstly introduced the fuzzification of binary operation to group structure through fuzzy equality [5] and introduced "vague groups." After this work, many researchers used this concept and extended it to several other useful directions such as [6–10]. In Demirci's approach, the characteristic of the degree between the fuzzy binary operation is not used, and the identity and inverse element of an element are also not unique. Liu and Shi [11] proposed a new approach to fuzzify the group structure by characterizing the degree of fuzzy binary operation, which is called *M*-hazy groups. It is important to mention that *M*-hazy associative law has been defined in order to obtain *M*-hazy groups. Mehmood et al. [12] extended this concept to the ring structure and gave a new method to the fuzzification of rings, which is defined by *M*-hazy rings. It is also worth mentioning that an *M*-hazy distributive law has been proposed so as to define *M*-hazy rings. Furthermore, Mehmood et al. [13] also provided the homomorphism theorems of *M*-hazy rings with its induced fuzzifying convexities. Liu and Shi [14] proposed *M*-hazy lattices. Fan et al. [15] introduced an *M*-hazy Γ-semigroup.

Vector space has been the most widely studied and used in linear algebra theory. A vector space is a set of elements with a binary addition operator and a multiplication operator that has closure under these two operations over a field, all while satisfying a set of axioms. Vector spaces are the realm of linear combinations, also known as superpositions, weighted sums, and sums with coefficients. Such sums occur throughout mathematics, both pure and applied, including statistics, science, engineering, and economics. The key word is "linear". Even when studying nonlinear phenomena, it is often useful to approximate with a simpler linear model. You can say that vector spaces are one of the grea<sup>t</sup> organizing tools of mathematics, helping reveal a structural similarity in a wide variety of topics found in such different contexts that they may seem completely different. Suppose you stand in front of a house. It is rather old but beautifully constructed of

**Citation:** Mehmood, F.; Shi, F.-G. *M*-Hazy Vector Spaces over *M*-Hazy Field. *Mathematics* **2021**, *9*, 1118. https://doi.org/10.3390/math9101118

Academic Editors: Sorin Nadaban and Ioan Dzitac

Received: 4 April 2021 Accepted: 12 May 2021 Published: 14 May 2021

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masonry that exhibits excellent craftsmanship. You point at a brick in one of the lower layers and ask "What is the need for this brick?" Short answer: it helps the structure. Long answer: it can be missed in the sense that the building won't fall apart if you take it away, but it will damage it in various ways that will become clear if you live there for a couple of years. This house is a metaphor for mathematics. A vector space is a lot more than just a brick. It is one of the fundamental notions, and so is part of the foundation. The most fundamental notion is "Set," and a vector space is one notch higher, a set with a specific structure. Can you do without it? Not if you want to do any serious mathematics. You can refine the structure to ge<sup>t</sup> topological vector space, metric vector space, complete vector space, normed vector space, and inner product vector space, each a refinement of the former. The beauty of this is that a refinement inherits all properties of its ancestor, so you are saved a lot of groundwork and can explore additional properties. The need for refinement is usually triggered by questions from physics or engineering. A nice example is Fourier analysis that fits smoothly in a Hilbert space structure.

Since Katsaras and Liu [16] presented the notion of fuzzy vector spaces, many scientists and researchers explored its properties and obtained fruitful research such as [17–21]. Mordeson [22] defined bases of fuzzy vector spaces. Shi and Huang [23] defined fuzzy bases and fuzzy dimension of fuzzy vector spaces. Nanda [24,25] introduced fuzzy fields and linear spaces. Malik and Mordeson [26] defined fuzzy subfield of a field. Fang and Yan [27] introduced the notion of *L*-fuzzy topological vector spaces. Zhang and Xu [28,29] presented the concept of topological *L*-fuzzifying neighborhood structures. Furthermore, Yan and Wu [30] extended this concept by introducing fuzzifying topological vector spaces on completely distributive lattices. Wen et al. [31] gave the degree to which an *L*-subset of a vector space is an *L*-convex set.

In the past few years, theory of convexity has emerged more and more important study for exceptional problems in many fields of applied mathematics. Since the 1950s, convexity theory has developed into several related theories. Van de Vel [32] conducted an inventive investigation. His work was praised as excellence. An interesting question about the application part of convex theory attempts to include determining the computational complexity of convexity, pattern recognition problems, optimization, etc. Fuzzy set theory is an emerging discipline in different fields of abstract algebra such as topology and convexity theory, etc. Rosa [33] firstly presented the notion of fuzzy convex spaces. Shi and Xiu [34] proposed a new technique for the fuzzification of convex structures, which is described as *M*-fuzzifying convex structures. In this technique, each subset of a set *X* has a certain degree of convexity. Furthermore, Shi and Xiu [35] gave the generalization of *L*-convex structure and *M*-fuzzifying convex structure by introducing (*<sup>L</sup>*, *M*)-fuzzy convex structure. In their approach, every *L*-fuzzy subset can be considered as an *L*-convex set to some extent. With repeated progress in the area of convexity theory, fuzzy convex structures have become a major research area such as [36–52]. Pang and Xiu [53] introduced the notion of lattice-valued interval operators and described their connection between *L*-fuzzifying convex structures. Liu and Shi [54] proposed *M*-fuzzifying median algebra, which is obtained through fuzzy binary operation. This study provided the characteristics of *M*-fuzzifying median algebra and *M*-fuzzifying convex spaces. This work provided motivation to extend it on more algebraic structures like groups, rings, lattices, and vector spaces. 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.

The paper is organized as follows: Section 2 consists of fundamental notions about completely residuated lattices, field and vector spaces, *M*-hazy groups, *M*-hazy rings, and *M*-fuzzifying convex spaces. In Section 3, the concept of *M*-hazy vector space is defined and obtained its fundamental properties. In Section 4, the concept of *M*-hazy subspaces is introduced, and it has been shown that all the *M*-hazy subspaces of *M*-hazy vector space form a convex structure. In Section 5, the linear transformation of *M*-hazy vector spaces is introduced. Finally, *M*-fuzzifying convex spaces are induced by *M*-hazy subspaces of *M*-hazy vector spaces. Section 6 concludes the paper.
