In this paper, two related quotient structures are investigated utilizing the concept of coset. At first, a new hypervector space F/V = (F/V,\circ,\circledcirc,K) is created, which is composed of all cosets of a bipolar fuzzy soft set (F;A) over a hypervector space V . Then it will be shown that dim F/V = dim V/W, where the quotient hypervector space V/W includes all cosets of an especial subhyperspace W of V. Also, three bipolar fuzzy soft sets over the quotient hypervector space V/W are presented and in this way some new bipolar fuzzy soft hypervector spaces are defined.

The aim of this paper is the study of fuzzy basis and dimension of fuzzy hypervector spaces. In this regard, first the notions of fuzzy linear independence and fuzzy basis are introduced and then some related results are obtained. In particular, it is shown that for a large class of fuzzy hypervector space the fuzzy basis exist. Finally, dimension of a fuzzy hypervector space is defined and the basic properties of that are investigated.

In this paper we investigate the algebraic properties of dimension of fuzzy hypervector spaces. Also, we prove that two isomorphic fuzzy hypervector spaces have the same dimension.

In this note first we shall redefine the notion of a fuzzy hypervector space (see [1]) and then we introduce some further concepts of fuzzy hypervector spaces, such as convex fuzzy sets and balance fuzzy sets in fuzzy hypervector spaces over valued fields. Finally, we briefly discuss on convex (balanced) hull of a given fuzzy set of a hypervector space.

The purpose of this paper is the study of sub-hyperspaces of a hypervector space investigating some of their properties. In this regard, the notion of a normal fuzzy sub-hyperspace of hypervector space is introduced. Using it, we construct new fuzzy sub-hyperspaces. We also show that, under certain conditions, a fuzzy sub-hyperspace of a hypervector space is two-valued and takes the values 0 and 1.

In this paper, we have generalized the definition of the vector space by considering the group as a canonical m-ary hypergroup, the field as a krasner (m; n)-hyperfield and considering the multiplication structure of a vector by a scalar as hyperstructure. Also we will be consider a normed m-ary hypervector space and introduce the concept of convergent of sequence on m-ary hypernormed spaces and bundle subset.

This article presents the exact solution of a bipolar fuzzy heat equation based on bipolar fuzzy Fourier transform under generalized Hukuhara partial (gH-p) differentiability. A bipolar fuzzy Fourier transform is defined, and the related key propositions and fundamental characteristics are discussed. Further, a bipolar fuzzy heat equation model is constructed using gH-differentiability, and the analytical solution of a bipolar fuzzy heat equation with bipolar fuzzy Fourier transform approach is examined. Some illustrative examples are provided to check the suggested methodology’s liability and efficiency. The type of differentiability and the solution of the bipolar fuzzy heat equation are shown graphically, demonstrating the versatility of the proposedmethodology and elucidating the impact of differentiability types on the solution behavior of the bipolar fuzzy heat equation. Additionally, the impact of different parameters on the solution behavior is analyzed, revealing insights into the underlying dynamics.

Molodtsov initiated soft set theory that provided a general mathematical framework for handling with uncertainties in which we encounter the data by affix parameterized factor during the information analysis as differentiated to fuzzy as well as bipolar fuzzy set theory. The main object of this paper is to lay a foundation for providing a new application of bipolar fuzzy soft tool in considering many problems that contain uncertainties. In present study, aggregation of bipolar fuzzy soft numbers have so far not yet been applied for ranking of alternatives in decision-making problems. To this propose, bipolar fuzzy soft weighted arithmetic averaging (BFSWAA) operator, bipolar fuzzy soft weighted geometric averaging (BFSWGA) operator have been used to compare two bipolar fuzzy soft numbers (BFSNs) for aggregating different bipolar fuzzy soft input arguments in bipolar fuzzy soft environment. Then, their related properties have been investigated. Finally, a practical example for selection of alternatives is given to demonstrate the utility and application of the proposed work.

The aim of this paper is to prove the Uniform Boundedness Principle and Banach-Steinhaus Theorem for anti linear operators and hence strong linear operators on Banach hypervector spaces. Also we prove the continuity of the product operation in such spaces.

In this paper, bipolar inverse fuzzy graphs are introduced. Several operations are made on such graphs and whether these operations result in a bipolar inverse fuzzy graph is investigated. A type of bipolar inverse fuzzy graphs called strong bipolar inverse fuzzy graphs have been studied. The complement graph and dependent threshold of a bipolar inverse fuzzy graph are introduced and it is concluded that the complement of a bipolar inverse fuzzy graph is a strong bipolar inverse fuzzy graph. An inclusion condition regarding threshold graphs corresponding to two bipolar inverse fuzzy graphs with special conditions will result.