Semiring of Generalized Interval-Valued Intuitionistic Fuzzy Matrices

Semiring of Generalized Interval-Valued Intuitionistic Fuzzy Matrices

Debashree Manna
Copyright: © 2017 |Pages: 21
DOI: 10.4018/978-1-5225-0914-1.ch006
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Abstract

In this paper, the concept of semiring of generalized interval-valued intuitionistic fuzzy matrices are introduced and have shown that the set of GIVIFMs forms a distributive lattice. Also, prove that the GIVIFMs form an generalized interval valued intuitionistic fuzzy algebra and vector space over [0, 1]. Some properties of GIVIFMs are studied using the definition of comparability of GIVIFMs.
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Preliminaries

Here some preliminaries, definitions of IVIFMs and GIVIFMs are recalled and presented some operations on GIVIFMs.

Definition 1

A semiring is an algebraic structure (R,+,.) such that (R,+) is an abelian monoid (identity 0), (R,.) is a monoid (identity 1). distributes over + from either side, r0=0r=0 for all r∈R and 0≠1.

Definition 2

A fuzzy matrix (FM) of order m×n is defined as 978-1-5225-0914-1.ch006.m01 where 978-1-5225-0914-1.ch006.m02 is the membership value of the ij-element in A. Let 978-1-5225-0914-1.ch006.m03 denote the set of all fuzzy matrices of order m×n. If m=n, in short, we write Fn, the set of all square matrices of order n.

Definition 3

An intuitionistic fuzzy matrix (IFM) of order m×n is defined as 978-1-5225-0914-1.ch006.m04 where aij, and 978-1-5225-0914-1.ch006.m05 are the membership value and non membership value of the ij-element in A satisfying the condition 978-1-5225-0914-1.ch006.m06 for all i,j.

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