Hypervector Spaces Research Articles

Hypernorm on hypervector spaces over a hyperfield

Hypernorm is a generalization of the notion of a norm on a vector space over a field. In this paper, we consider a hypervector space (V,+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mathbb {V}, +)$$\\end{document} over a hyperfield, where +\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$+$$\\end{document} is a hyperoperation, and prove that the hypernorm is continuous. We show that the natural linear transformation from V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {V}$$\\end{document} to VZ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dfrac{\\mathbb {V}}{Z}$$\\end{document} is continuous and open for all closed subhyperspaces Z of V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {V}$$\\end{document}. We prove BL(V,W),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$BL(\\mathbb {V},\\mathbb {W}),$$\\end{document} the set of all bounded linear transformations from V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {V}$$\\end{document} to W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {W}$$\\end{document} is a hyper-Banach space whenever W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {W}$$\\end{document} is complete. Furthermore, we obtain that in a hyper-Banach space V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {V}$$\\end{document} if {μn}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lbrace \\mu _n \\rbrace$$\\end{document} is a sequence of continuous linear transformations with {/μn(u)/}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lbrace /\\mu _n(u)/ \\rbrace$$\\end{document} is bounded for every u∈V,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u \\in \\mathbb {V},$$\\end{document} then {‖μn‖}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lbrace \\Vert \\mu _n\\Vert \\rbrace$$\\end{document} is bounded. In the sequel, we prove several properties of hypernorm and linear transformations on hypernormed spaces.

Retracted: Roughness in Hypervector Spaces

Retracted: Roughness in Hypervector Spaces

Topological Vector Spaces Derived From Topological Hypervector Spaces

We introduce topological hypervector spaces on a topological field, in the sense of Tallini, and study some basic properties of this hyperspaces. In this regards we study the relationship between the topology on a hypervector spaces and its complete part. In particular we show that if every open subset of a topological hypervector space is a complete part then its fundamental vector space induced is a topological vector space. Finally, we study the quotient space of topological hypervector spaces and the derived topological space of a topological hypervector space with respect its fundamental relegation.

Quotient bipolar fuzzy soft sets of hypervector spaces and bipolar fuzzy soft sets of quotient hypervector spaces

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.

Roughness in Hypervector Spaces

This paper examines rough sets in hypervector spaces and provides a few examples and results in this regard. We also investigate the congruence relations-based unification of rough set theory in hypervector spaces. We introduce the concepts of lower and upper approximations in hypervector spaces.

Picture fuzzy sub-hyperspace of a hyper vector space and its application in decision making problem

<abstract><p>In this paper, the notion of picture fuzzy sub-hyperspace of a hyper vector space is introduced and some related results are investigated on the basis of some basic operations (intersection, union, Cartesian product etc.) on picture fuzzy sets. The concept of picture fuzzy linear transformation with respect to some picture fuzzy sub-hyperspace is initiated here and some important results are studied in this regard. It is shown that with respect to some pre-assumed picture fuzzy sub-hyperspace, linear combination of two picture fuzzy linear transformations is a picture fuzzy linear transformation, composition of two picture fuzzy linear transformations is a picture fuzzy linear transformation and inverse of a bijective picture fuzzy linear transformation is a picture fuzzy linear transformation. The effect of good linear transformation on picture fuzzy sub-hyperspaces is discussed here. It is shown that the image of a picture fuzzy sub-hyperspace is a picture fuzzy sub-hyperspace under bijective good linear transformation and the inverse image of a picture fuzzy sub-hyperspace is a picture fuzzy sub-hyperspace under good linear transformation. Some important results on picture fuzzy sub-hyperspaces in the light of $ (\theta, \phi, \psi) $-cut of picture fuzzy set are studied here. Finally, an application of picture fuzzy sub-hyperspace conditions in decision making problem is presented here.</p></abstract>

Affine and convex fuzzy subsets of hypervector spaces

Affine and convex fuzzy subsets of hypervector spaces

Intersectional soft sets theory applied to generalized hypervector spaces

Abstract The notion of θ-generalized int-soft subfields, θ-generalized int-soft algebras over θ-generalized int-soft subfields, and θ-generalized int-soft hypervector spaces are introduced, and their properties and characterizations are considered. In connection with linear transformations, θ-generalized int-soft hypervector spaces are discussed.

Hyper vector spaces over Krasner hyperfields

In this paper, we study hyper vector spaces over Krasner hyperfields. First, we introduce the notions of linearly independence (dependence) and basis for a hyper vector space. Second, we investigate their properties and prove some results for hyper vector spaces that are similar to that of vector spaces over fields. Then, we define linear transformations over hyper vector spaces and investigate their properties. Finally, we prove the dimension theorem for linear transformations.

Hyperoperations defined on sets of S -helix matrices

A hyperproduct on non-square ordinary matrices can be defined by using the helix-hyperoperation. Therefore, the helix-hyperoperation (abbreviated hope ) is based on a classical operation and was introduced in order to overcome the non-existing cases. We study the helixhyperstructures on the special type of matrices, the Shelix matrices, used on the small dimension representations. In this paper, we introduce and focus our study on the class of S-helix matrices called k-overlap helix matrices. The reason is that their hyper-vector spaces can represent n-dimensional spaces which have independent both, single valued dimensions and multivalued dimensions.

On Refined Neutrosophic Hypervector Spaces

On Refined Neutrosophic Hypervector Spaces

Additive Maps Preserving the Essential Points Between Weak Hypervector Spaces

IIn this paper, we introduce the concept of regular linear operators. We prove that the regular linear operators preserve the normality, linear independence and dimension of weak hypervector spaces and prove some important results.

Balanced and absorbing fuzzy subsets of hypervector spaces

In this paper, the notion of a balanced subset of a hypervector space over the field of real numbers is generalized to a hypervector space over an arbitrary valued field. Moreover, absorbing subsets of hypervector spaces are redefined. Next, the concept of balanced fuzzy subsets of hypervector spaces over valued fields is redefined and the balanced hull of fuzzy subsets of hypervector spaces is presented. Also, the notion of absorbing fuzzy subsets of hypervector spaces is introduced and some related results are obtained. Finally, the behavior of the mentioned notions under linear transformations is investigated.

Linear functionals on hypervector spaces

The study of linear functionals, as an important special case of linear transformations, is one of the key topics in linear algebra and plays a significant role in analysis. In this paper we generalize the crucial results from the classical theory and study main properties of linear functionals on hypervector spaces. In this way, we obtain the dual basis of a given basis for a finite-dimensional hypervector space. Moreover, we investigate the relation between linear functionals and subhyperspaces and conclude the dimension of the vector space of all linear functionals over a hypervector space, the dimension of sum of two subhyperspaces and the dimension of the annihilator of a subhyperspace, under special conditions. Also, we show that every superhyperspace is the kernel of a linear functional. Finally, we check out whether every basis for the vector space of all linear functionals over a hypervector space V is the dual of some basis for V.

On Neutrosophic Quadruple Hypervector Spaces

The objective of this paper is to study Neutrosophic Quadruple Hypervector Spaces and present some of their basic definitions and properties. This paper generalizes the concept of Neutrosophic Hypervector spaces by presenting their Neutrosophic Quadruple forms. Some notions such as Neutrosphic hypersubspaces, linear combination, linearly dependence and linearly independence are generalized. Some interesting results and examples to illustrate the new concepts introduced are presented.

On Refined Neutrosophic Hypervector Spaces

This paper presents the refinement of neutrosophic hypervector spaces and studies some of its basic properties. Some basic definitions and important results are presented. The paper also establishes the existence of a good linear transformation between a weak refined neutrosophic hypervector space V (I1; I2) and a weak neutrosophic hypervector space V (I).

Sum and scalar product of fuzzy subhyperspaces

The aim of this paper is to study the properties of two operations, sum and scalar product, on fuzzy subsets of hypervector spaces. In this regards, some new equivalent conditions of fuzzy subhyperspaces are introduced based on these operations. Also, the behavior of sum and scalar product of fuzzy subhyperspaces under linear transformations are investigated.

Soft Hypervector Space and Fuzzy Soft Hypervector Space

In this paper, the notions of soft hypervector space and fuzzy soft hypervector space are introduced, and several basic properties are provided. Also, we define and analyze the concept of image and pre-image of fuzzy soft hypervector space.

Various kinds of fuzzy quotient hypervector spaces

Various kinds of fuzzy quotient hypervector spaces

Hypervector Spaces Based on Intersectional Soft Sets

The notion of int-soft subfields, int-soft algebras over int-soft subfields, and int-soft hypervector spaces are introduced, and their properties and characterizations are considered. In connection with linear transformations, int-soft hypervector spaces are discussed.