Countable Fuzzy Topological Space and Countable Fuzzy Topological Vector Space

This paper deals with countable fuzzy topological spaces, a generalization of the notion of fuzzy topological spaces. A collection of fuzzy sets F on a universe X forms a countable fuzzy topology if in the definition of a fuzzy topology, the condition of arbitrary supremum is relaxed to countable supremum. In this generalized fuzzy structure, the continuity of fuzzy functions and some other related properties are studied. Also the class of countable fuzzy topological vector spaces as a generalization of the class of fuzzy topological vector spaces has been introduced and investigated

Set Linear Algebra and Set Fuzzy Linear Algebra

In this book, the authors define the new notion of set vector spaces which is the most generalized form of vector spaces. Set vector spaces make use of the least number of algebraic operations, therefore, even a non-mathematician is comfortable working with it. It is with the passage of time, that we can think of set linear algebras as a paradigm shift from linear algebras. Here, the authors have also given the fuzzy parallels of these new classes of set linear algebras. This book is divided into seven chapters. The first chapter briefly recalls some of the basic concepts in order to make this book self-contained. Chapter two introduces the notion of set vector spaces which is the most generalized concept of vector spaces. Set vector spaces lends itself to define new classes of vector spaces like semigroup vector spaces and group vector spaces. These are also generalization of vector spaces. The fuzzy analogue of these concepts are given in Chapter three. In Chapter four, set vector spaces are generalized to biset bivector spaces and not set vector spaces. This is done taking into account the advanced information technology age in which we live. As mathematicians, we have to realize that our computer-dominated world needs special types of sets and algebraic structures. Set n-vector spaces and their generalizations are carried out in Chapter five. Fuzzy n-set vector spaces are introduced in the sixth chapter. The seventh chapter suggests more than three hundred problems.Comment: 344 page

On properties of fuzzy subspaces of vectorspaces

In this paper, we introduce the notion of normal fuzzy subspace of vector spaces. By using it, we construct new fuzzy subspaces. We also show that, under certain conditions, a fuzzy subspace of a vector space is two-valued and takes 0 and 1

Fuzzy bases and the fuzzy dimension of fuzzy vector spaces

In this paper, new definitions of a fuzzy basis and a fuzzy dimension for a fuzzy vector space are presented. A fuzzy basis for a fuzzy vector space $(E,mu)$ is a fuzzy set $beta$ on $E$. The cardinality of a fuzzy basis $beta$ is called the fuzzy dimension of $(E,mu)$. The fuzzy dimension of a finite dimensional fuzzy vector space is a fuzzy natural number. For a fuzzy vector space, any two fuzzy bases have the same cardinality. If $widetilde{E}_1$ and $widetilde{E}_2$ are two fuzzy vector spaces, then $dim(widetilde{E}_1+widetilde{E}_2)+dim(widetilde{E}_1cap widetilde{E}_2)=dim(widetilde{E}_1) +dim(widetilde{E}_2)$ and $dim({widetilde{rm{ker }}f})+dim({widetilde{rm{im }}f})=dim(widetilde{E})$ hold without any restricted conditions. end{abstract

Fuzzy Linguistic Topological Spaces

This book has five chapters. Chapter one is introductory in nature. Fuzzy linguistic spaces are introduced in chapter two. Fuzzy linguistic vector spaces are introduced in chapter three. Chapter four introduces fuzzy linguistic models. The final chapter suggests over 100 problems and some of them are at research level.Comment: 193 pages; Published by Zip publishing in 201