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Intuitionistic fuzzy matrices
... Fuzzy matrices are extremely useful in dealing with large amount of information such as Agriculture, Medicine and any other field dealing with uncertainty in information. Pal et al. [14] have introduced the concept of intuitionistic fuzzy matrices as an extension to the theory of ordinary fuzzy matrices, which is the hybridization of intuitionistic fuzzy set and matrix theory. ...
... Definition 2.2 (Boolean Intuitionistic Fuzzy Matrix [14]). An intuitionistic fuzzy matrix P * = [ η P * (α i j ), θ P * (α i j )] m×n is said to be Boolean if its all elements are either 0 or 1. [14]). ...
... Definition 2.2 (Boolean Intuitionistic Fuzzy Matrix [14]). An intuitionistic fuzzy matrix P * = [ η P * (α i j ), θ P * (α i j )] m×n is said to be Boolean if its all elements are either 0 or 1. [14]). An intuitionistic fuzzy matrix P * = [ η P * (α i j ), θ P * (α i j )] m×n is said to be Boolean if it's all elements is 0.5. ...
For solving multi-criterion decision making problems, we in this paper propose a parametric generalized logarithmic divergence measure for intuitionistic fuzzy matrices. The validity of a symmetric divergence measure has been established for the proposed measure. Also, the properties (compliment, transitivity, concavity and symmetricity) of this measure for intuitionistic fuzzy matrices are studied. Application of the proposed measure has been illustrated through a decision-making problem in trade market.
... The concept of intuitionistic fuzzy matrices was introduced by Pal et al. [1] as a generalization of the well known ordinary fuzzy matrices introduced by Thomason [2] which take its elements from the unit interval [0,1]. An intuitionistic fuzzy matrix is a pair of fuzzy matrices, namely, a membership and non-membership function which represent positive and negative aspects of the given information (see [3,4] ). ...
... The concept of intuitionistic fuzzy matrices was introduced by Pal et al. [1] as a generalization of the well known ordinary fuzzy matrices introduced by Thomason [2] which take its elements from the unit interval [0,1]. An intuitionistic fuzzy matrix is a pair of fuzzy matrices, namely, a membership and non-membership function which represent positive and negative aspects of the given information (see [3,4] ). ...
... We may write 0 instead of < 0, 1 > and 1 instead of < 1, 0 > . [1,3,[8][9][10][11]. For an n × n intuitionistic fuzzy matrix A we have: ...
In this paper, we define the intuitionistic circular fuzzy matrix and introduce the necessary and sufficient conditions for an intuitionistic fuzzy matrix to be circular. Also, we study some properties of intuitionistic circular fuzzy matrices
... Im et al. [13], studied the determinant of square intuitionistic fuzzy matrices. Pal et al. [14], developed the intuitionistic fuzzy matrices and studied several properties of it using the idea of IFSs and intuitionistic fuzzy determinant. Later, Shyamal and Pal [15] studied some more properties on fuzzy matrices. ...
... The notion of IFM was given by Atanassov [28]. Further, the concept of IFM was developed by Pal et al. [14] have some special features which are not available in fuzzy matrix theory . Normally fuzzy matrix theory deals only with relevant information, but IFM deals with both relevant and irrelevant information. ...
... [29] An Intuitionistic Fuzzy Set (IFS) A in X, where X denotes a universal set is defined as an object of the following form A = {{x, µ A (x), ν A (x)/x ∈ X}, where the functions: µ A : X → [0, 1] and ν A : X → [0, 1] define the membership function and non-membership function of the element x ∈ X respectively and for every x ∈ X : 0 ≤ µ A (x) + ν A (x) ≤ 1. Definition 2.2. [14] Let X = {x 1 , x 2 , ...x m } be a set of alternatives and Y = {y 1 , y 2 , ...y n } be the attribute set of each element of X. An IFM is defined by A = ((x i , y j ), µ A (x i , y j ), ν A (x i , y j )) for i = 1, 2...m and j = 1, 2, ...n, where µ A : X × Y → [0, 1] and ν A : X × Y → [0, 1] satisfy the condition 0 ≤ µ A (x i , y j ) + ν A (x i , y j ) ≤ 1. ...
In this paper, we examine idempotent intuitionistic fuzzy matrices and idempotent intuitionistic fuzzy matrices of T-type. We develop some properties on both idempotent intuitionistic fuzzy matrices and idempotent intuitionistic fuzzy matrices of T-type. Several theorems are provided and an numerical example is given to illustrate the theorems.
... Definition 2.1 ( [12,15]). An intuitionistic fuzzy matrix(IFM) is a matrix of pairs A = ( µ aij , ν aij )of a non negative real numbers satisfying µ aij + ν aij ≤ 1 for all i, j. 16]). ...
... Definition 2.7 ( [15]). Let A = ( µ aij , ν aij ) and B = ( µ bij , ν bij ) be two intuitionistic fuzzy matrices. ...
In this paper, we study the algebraic sum and algebraic product of intuitionistic fuzzy matrices and prove that the set of all intu-itionistic fuzzy matrices forms a commutative monoid. We prove that the DeMorgan's laws of intuitionistic fuzzy matrices and we also prove that the distributive laws of intuitionistic fuzzy matrices are satisfied. 2010 AMS Classification: 03E72, 15B15, 15B99
... generalizes fuzzy matrix as IFM in [6,7] and they studied the determinant and adjoint of square IFMs. Khan et al. established the same in [8] which has been useful in dealing with the areas such as decision making, relational equations, clustering analysis etc. Intuitionistic fuzzy algebra and its matrix theory are considered by several researchers in using component wise max-min and min-max operations in various years as follows. ...
... Definition 2.2. [8,9] An intuitionistic fuzzy matrix A = [(a i j , a i j )] m×n be a matrix where a i j and a i j are the membership value and non membership value of the i j th element of A satisfying the condition that 0 ≤ a i j + a i j ≤ 1 for all i, j as well as its operations are defined as follows ...
In this article, different kinds of component wise max-max operations are introduced on intuitionistic fuzzy matrix
and some algebraic properties are studied together with the component wise min-min(^m) operation . Also some
semigroup algebraic structures are constructed using these new operations on intuitionistic fuzzy matrices.
... generalizes fuzzy matrix as IFM in [6,7] and they studied the determinant and adjoint of square IFMs. Khan et al. established the same in [8] which has been useful in dealing with the areas such as decision making, relational equations, clustering analysis etc. Intuitionistic fuzzy algebra and its matrix theory are considered by several researchers in using component wise max-min and min-max operations in various years as follows. ...
... Definition 2.2. [8,9] An intuitionistic fuzzy matrix A = [(a i j , a i j )] m×n be a matrix where a i j and a i j are the membership value and non membership value of the i j th element of A satisfying the condition that 0 ≤ a i j + a i j ≤ 1 for all i, j as well as its operations are defined as follows ...
In this article, different kinds of component wise max-max operations are introduced on intuitionistic fuzzy matrix and some algebraic properties are studied together with the component wise min-min (∧ m) operation. Also some semigroup algebraic structures are constructed using these new operations on intuitionistic fuzzy matrices.
... The index matrix representation of the intuitionistic fuzzy graphs has been studied in Atanssov (1994). Pal et al. (2002), Meenakshi and Gandhimathi (2010), and Murugadas (2011, 2010) studied IFM for finding intuitionistic fuzzy linear relation equation, g-inverse and intuitionistic fuzzy linear transformation and others. Meenakshi (2008) studied minus ordering, space ordering and schur complement of fuzzy matrix and block fuzzy matrix. ...
... Definition 2.1. (Pal et al. (2002)) Let X = {x 1 , x 2 , ...x m } be a set of alternatives and Y = {y 1 , y 2 , ...y n } be the attribute set of each element of X. An Intuitionistic Fuzzy Matrix (IFM) is defined by ...
A problem of reducing intuitionistic fuzzy matrices is examined and some useful properties are obtained with respect to nilpotent intutionistic fuzzy matrices. First, reduction of irreflexive and transitive intuitionistic fuzzy matrices are considered, and then the properties are applied to nilpotent intutionistic fuzzy matrices. Nilpotent intuitionistic fuzzy matrices are intuitionistic fuzzy matrices which signify acyclic graphs, and the graphs are used to characterize consistent systems. The properties are handy for generalization of various systems with intuitionistic fuzzy transitivity.
... Further every invertible matrix is regular. Thus regular fuzzy matrices play an important role in estimation and inverse problem in fuzzy relational equations [10, PP: [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and in fuzzy optimization problem [9,. For more details on fuzzy matrices one may refer [6]. ...
... Atanassov has introduced and developed the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets [1,2]. Basic properties of intuitionistic fuzzy matrices as a generalization of the results on fuzzy matrices have been derived by Pal et.al [7]. The generalized inverse of intuitionistic fuzzy matrices was discussed in [5]. ...
... Pal and Khan [4] introduced intuitionistic fuzzy tautological matrices and its algebraic operations. Murugadas and Lalitha [5] they defined intuitionistic fuzzy cotautological matrices and its algebraic operations. ...
In this paper, introduced Pythagorean fuzzy tautologial matrices and Pythagorean fuzzy cotautological matrices and some properties of Lukasiwicz implication operator over Pythagorean fuzzy tautologial matrices and Pythagorean fuzzy cotautological matrices are discussed. Also discussed the relation between implication with Lukasiewicz disjunction and conjunction operations of PFCMs and PFCTMs.
... Im et al. (2001), Pal (2001), and Khan et al. (2002) generalize a fuzzy matrix as IFM in with its operations and has been useful in dealing with the areas such as decision making, relational equations, clustering analysis, etc. IFM is also very useful in the discussion of Intuitionistic fuzzy relation. Pal ((2002-2003), (2005)) introduced intuitionistic fuzzy tautological matrix (IFTM) and interval value intuitionistic fuzzy matrices. Later, Murugadas and Lalitha (2015) developed intuitionistic fuzzy cotautological matrix (IFCTM). ...
In this article, we extend the mathematical operation symmetrical difference () to intuitionistic fuzzy matrix. Various properties of the difference operator ' ' are discussed over intuitionistic fuzzy matrices. Also associativity property of the above said operator is studied when each entry of an intuitionistic fuzzy matrix is either intuitionistic fuzzy tautological or co tautological since it is very critical when we prove that in usual manner. Finally, a commutative monoid algebraic structure is obtained on symmetrical difference operator over intuitionistic fuzzy matrices.
... Im et.al.(2001) and Khan et al.(2002) generalizes fuzzy matrix as IFM in with its operations and has been useful in dealing with the areas such as decision making, relational equations, clustering analysis etc. IFM is also very useful in the discussion of Intuitionistic fuzzy relation. Pal(2002-2003) introduced 2 xxx intuitionistic fuzzy tautological matrix(IFTM) and then Murugadas and Lalitha(2015) developed intuitionistic fuzzy cotautological matrix(IFCTM). A lot of research activities with several algebraic structures have been carried out during various years by different researchers on IFMs in (Im et al.(2003), Boobalan and Sriram(2016), Muthuraji and Sriram(2016), Atanassov(2017), Muthuraji and Sriram(2017), Silambaran and Sriram(2019), Silambarasan and Sriram(2020)). ...
In this article, we extend the mathematical operation symmetrical difference (⊖) to intuitionistic fuzzy matrix. Various properties of the difference operator '⊖' are discussed over intutionistic fuzzy matrices. Also associativity property of the above said operator is studied when each entry of an intuitionistic fuzzy matrix is either intuitionistic fuzzy tautological or co tautological since it is very critical when we prove that in usual manner. Finally a commutative monoid algebraic structure is obtained on symmetrical difference operator over intuitionistic fuzzy matrices.
... The min max composition of fuzzy matrices have been studied by Ragab and Emam [11]. Atanassov has introduced and developed the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets [1,2].The intuitionistic fuzzy matrices as a generalization of the results on fuzzy matrices have been discussed in [9]. Basic properties of intuitionistic fuzzy matrices under Cartesian product representation has discussed in [3]. ...
... S.K.Khan and M.Pal initiated Inituitionistic fuzzy tautological matrix(IFTM), a varient of IFM with an implication operator in [5]. Similar to this, the authors in [6] developed another kind of implication with various properties of Intuitionistic fuzzy cotautological matrix(IFCTM). ...
In this paper, various properties of Lukasiwicz implication operator over intuitionistic fuzzy tautologial matrices and intuitionistic fuzzy cotautological matrices are studied. Also some expressions which involves Lukasiwicz conjunction, disjunction and implication operators on intuitionistic fuzzy matrix are discussed.
... an IFM and discuss some properties. Ronald R. Yager [11], Im et al [12], Khan S. K, Pal M and Amiya K. Shyamal [3], Meenakshi A. R and Gandhimathi [4], Sriram and Murugadas [6] and several authors discussed Intuitionistic Fuzzy Matrices. In [14] Wang et.al developed a new approach to constructing an intuitionistic fuzzy similarity matrix based on IFM. ...
In this paper, in IFM we introduce two commutative monoids using the operators namely Luckasiwicz conjuntion and disjuntion operators ⊕ and. In addition we discuss some properties like reflexive, symmetric, associative of the operators. Also monoid homomorphism has been defined.
... Keeping this type of situation in consideration, to removing limitations, intuitionistic fuzzy set (IFS) was proposed by Atanassov [2]. Later on, a lot of works on intuitionistic fuzzy matrices (IFMs) were done by different researchers [6,7,10,12,14]. The role of similarity is widely analyse by Cross et al. [3]. They insist the fundamental preface of ability and similarity in hypothesis and in applications in inferential argument using concept of FS theory. ...
... However, in the modeling of some problems involving uncertain data classical matrix theory may not be sufficient. Therefore, many researcher studied on matrix structures under fuzzy environment [16,29,32], intuitionistic fuzzy environment [13,24,25], soft environment [4], fuzzy soft environment [5] and intuitionistic fuzzy soft environment [9,22]. As we know, fuzzy set and intuitionistic fuzzy sets are important tools for dealing with problems containing uncertainty and incomplete information. ...
The single-valued neutrosophic set plays a crucial role to handle indeterminant and inconsistent information during decision making process. In recent research, a development in neutrosophic theory is emerged, called single-valued neutrosophic matrices, are used to address uncertainties. The beauty of single-valued neutrosophic matrices is that the utilizing of several fruitful operations in decision making. In this paper, some novel operations on neutrosophic matrices of are introduced, that is, type-1 product ( ˜ ?), type-2 product ( ˜ ⊗) and minus ( ˜ ?) between two single-valued neutrosophic matrices. Also, we introduced complement, transpose, upper and lower α−level matrices of single-valued neutrosophic matrices and discussed related properties. Furthermore, we propose a multi-criteria group decision making method based on these new operations,
and give an application of the proposed method in a real life problem. Finally, we compare proposed method in this paper
with proposed methods previously.
... Pal, et al., [26] studied intuitionistic fuzzy matrices (IFMs). Khan and Pal [27] studied intuitionistic fuzzy tautological matrices and also studied interval-valued IFMs [28]. Bhowmik and Pal [29,30] introduced some results on IFMs and intuitionistic circulant FMs and GIFMs. ...
In this paper, generalized intuitionistic fuzzy matrices are considered as matrices over a special type of semiring which is called path algebra. We introduce the concept of transitivity of generalized intuitionistic fuzzy matrices. Some algebraic properties of generalized intuitionistic fuzzy matrices are developed. Also, we develop some properties of transitivity.
... Simultaneously, Pal et al. [8] defined the IFM and Pal [15] introduced the intuitionistic fuzzy Determinant, studied some properties on it. Khan and Pal [9] studied some operations on IFMs. ...
... Pal, et al., [19] studied intuitionistic fuzzy matrices (IFMs). Khan and Pal [20] studied on intuitionistic fuzzy tautological matrices and also studied interval-valued IFMs [21]. Bhowmik and Pal [22,23] introduced some results on IFMs and intuitionistic circulant FMs and GIFMs. ...
In this paper, intuitionistic fuzzy matrices are considered. Szpilrajn's theorem on orderings is generalized to intuitionistic fuzzy orderings by Zedam et al. We give another generalization in intuitionistic fuzzy matrix form and theorem is represented in terms of intuitionistic fuzzy matrix operations.
... Definition 2.2 [5] Let X = (x1, x2,..., xm) be a set of alternatives and Y = (y1, y2,..., yn) be the attribute set of each element of X. An IFM is defined by ...
Convergences of powers of controllable intuitionistic fuzzy matrices have been studied. It is shown that they oscillate with period equal to 2, in general. Some equalities and sequences of inequalities about powers of controllable intuitionistic fuzzy matrices have been obtained.
... Powers and nilpotent conditions of matrices over a distributive lattice are consider by Tan [41]. After that Pal, Bhowmik, Adak, Shyamal, Mondal have done lot of works on fuzzy, intuitionistic fuzzy, interval-valued fuzzy, etc. matrices [1][2][3][4][5][6][7][8][9][10][11][12][25][26][27][28][29][30][31][32][35][36][37][38][39]. ...
In this paper, we define two new type of operators of fuzzy matrices denoted by the symbol Å and Ä.Using these operators of fuzzy matrices we define row-maxaverage norm, column-max-average norm. Here instead of addition of fuzzy matrices we use the operator Å and instead of multiplication of fuzzy matrices we use the operatorÄ.We also define Pseudo norm of fuzzy matrices and max-min norm.
... The notion of a triangular fuzzy matrix was proposed for the first time by Shyamal and Pal (Shayamal and Pal, 2007 ) and was made familiar through introducing some new operators on triangular fuzzy matrices (Shayamal and Pal, 2004). The progression of fuzzy numbers became so fruitful that it spread into intuitionistic fuzzy matrices (Adak et al., 2012a; Adak et al., 2012b; Bhowmik and Pal, 2012; Mondal and Pal, 2014; Pal, 2001; Pradhan and Pal, 2014a; Pradhan and Pal, 2014b; Pradhan and Pal, 2012; Shayamal and Pal, 2002) and interval valued fuzzy set theory (Mondal and Pal, 2015; Pal and Khan, 2005; Shayamal and Pal, 2006). In this article, we introduce the notion of pentagonal fuzzy number in a well-defined manner by generalizing some other types of fuzzy numbers and studied the basic arithmetic and algebraic properties of the pentagonal fuzzy number. ...
In this article, the notion of pentagonal fuzzy number (PFN) is introduced in a generalized way. A few articles have been published based on this topic, but they have some ambiguities in defining this type of fuzzy number. Here, we proposed the logical definition in developing a pentagonal fuzzy number, along with its arithmetic operations. Based on PFN, the structure of pentagonal fuzzy matrices (PFMs) is studied, together with their basic properties. Some special type of PFMs and their algebraic natures (trace of PFM, adjoint of PFM, determinant of PFM, etc.) are discussed in this article. Finally, the notion of nilpotent PFM, comparable PFM, and constant PFMs, with their many properties, are highlighted in this article.
... To overcome these difficulties, Atanassov [4] introduced theory of intuitionistic fuzzy set in 1983 as a generalization of fuzzy set. Based on this concept Pal et al. have defined intuitionistic fuzzy determinant in 2001 [39] and intuitionistic fuzzy matrices (IFMs) in 2002 [40]. Bhowmik and Pal [6][7][8][9][10]introduced some results on IFMs, intuitionistic circulant fuzzy matrix and generalized intuitionistic fuzzy matrix. ...
Eigenvalues and eigenvectors are one of the important topics over bipolar fuzzy linear algebra. In order to develop the bipolar fuzzy linear space we introduce in this article, the similarity relations, eigenvalues and eigenvectors of bipolar fuzzy matrices (BFMs). Idempotent, diagonally dominant and spectral radius of BFMs are considered here. Also, some properties and results of eigenvalues and eigenvectors for BFMs are investigated.
... Shyamal et al. [1] studied distance of intuitionistic fuzzy set and discussed interval valued intuitionistic fuzzy set. Some further studies in this direction can be seen in [13][14][15][16][17][18][19][20][21]. Shinoj and Sunil [8] proposed the concept of intuitionistic fuzzy multisets (IFMSs); theoretical views of IFMSs and some applications were given in [4][5][6][7][9][10]. ...
In this paper we proposed some new operations on intuitionistic fuzzy multisets (IFMSs), deduced some theorems with respect to the algebra of IFMSs and modal operators on IFMSs.
... First time Pal [15] introduced intuitionistic fuzzy determinant. Latter Pal and Shyamal [14, 17] introduced intuitionistic fuzzy matrices and distance between intuitionistic fuzzy matrices. Meenakshi and Jenita [12, 13] studied on − k regularity of block fuzzy matrix and Schur complement in − k kernel symmetric matrices. ...
In this paper we define multiplication between intuitionistic fuzzy matrices (IFMs) and we
derive the conditions for a block IFM to be regular. Also a method to find the generalized
inverse of it with the help of the generalized inverses of the blocks of the original matrix is
described. Again, it is shown that a block intuitionistic fuzzy matrix can be decomposed into
an upper triangular idempotent intuitionistic fuzzy matrix and a lower triangular idempotent
intuitionistic fuzzy matrix when the decomposition is symmetric.
... The fuzzy matrix have been proposed to represent fuzzy relation in a system based on fuzzy sets theory. Several authors presented a number of results on fuzzy matrices 3,4,5,6,7,8 . Pal and Shyamal 9,10 shown several properties on fuzzy matrices and interval-valued fuzzy matrices. ...
In this paper, a method is presented to find generalized inverse (g-inverse) of Atanassov's intuitionistic fuzzy matrix (AIFM). Also, the idea of standard basis for Atanassov's intuitionistic fuzzy vectors is introduced. Some results regarding the g-inverse of AIFM are studied. An application of g-inverse is provided at the end of the paper.
... The fuzzy matrix have been proposed to represent fuzzy relation in a system based on fuzzy sets theory. Several authors presented a number of results on fuzzy matrices 3,4,5,6,7,8 . Pal and Shyamal 9,10 shown several properties on fuzzy matrices and interval-valued fuzzy matrices. ...
In this paper, a method is presented to find generalized inverse (g-inverse) of Atanassov's intuitionistic fuzzy matrix (AIFM). Also, the idea of standard basis for Atanassov's intuitionistic fuzzy vectors is introduced. Some results regarding the g-inverse of AIFM are studied. An application of g-inverse is provided at the end of the paper.
... In [3], Thomason has introduced the concept of fuzzy matrices. After that a lot of works have been done on fuzzy matrices and its variants [4,5,6]. It is well known that the membership value completely depends on the decision maker's, its habit, mentality etc. ...
In this paper, we introduced interval-valued fuzzy matrices (IVFMs) as the generalization of interval-valued fuzzy sets. Some essential unary and binary operations of IVFM and some special types of IVFMs i.e., symmetric, re-flexive, transitive and idempotent, con-stant, etc. are defined here. The idea of convergence, periodicity, determinant and adjoint of IVFMs are also defined. Lot of properties of IVFMs are pre-sented here.
The collection of s-transitive and w-transitive intuitionistic fuzzy matrices comprise properly the collection of the transitive intuitionistic fuzzy matrices for which reduction models have already been proved. We have proved that basic properties of these models also holds for s-transitive and w-transitive intuitionistic fuzzy matrices Keywords: Intuitionistic Fuzzy Matrix, s-transitive intuitionistic fuzzy matrix , w-transitive intuitionistic fuzzy matrix.
Suppose there is a controversial statement say S. It is assumed that in a locality there are N people among them m persons accepting the statement S and n number of persons making negative comments about S, where . Then the ratios are called the degree of acceptance and degree of negation of the statement S. If , then everybody clearly mentioned his/her opinion and the situation can be handled with the fuzzy set (FS) only. If , then some people are not participated to make any comments about the statement S.
The use of matrix theory in decision-making problems has been a subject of great interest for researchers. However, recent developments have shown that matrices can be enhanced by incorporating fuzzy, intuitionistic fuzzy, and picture-fuzzy theory. Inspired by the notion of interval-valued picture fuzzy sets (IVPFSs), we extend the idea of picture fuzzy matrix (PFM) into interval-valued picture fuzzy matrix (IVPFM) to represent more flexibly uncertain and vague information. The paper defines several key definitions and theorems for the IVPFM and presents a procedure for calculating its determinant and adjoint. Using composition functions, we develop algorithms to identify the greatest and least eigenvalue interval-valued picture fuzzy sets and use a flow chart to illustrate the procedure. The work demonstrates this process with a numerical example of a decision-making problem. In addition, we introduce a new distance measure for the IVPFSs and prove its validity with the help of basic properties. Further, the application of the proposed concepts has been shown by a real-life numerical example of a computer numerical controlled (CNC) programmer selection problem in a smart manufacturing company.
In this article implication operations (→, ←) have been introduced for general picture fuzzy matrix(PFM). Sub inverses and g-inverses of PFM are obtained using the terminologies given in [20]. Further relations such as reflexive, transitive and idempotent are examined.
In this paper, the notions of picture fuzzy matrix, restricted picture fuzzy matrix and special restricted picture fuzzy matrix are established. Two types of -cut of special restricted square picture fuzzy matrix are introduced and corresponding properties are studied. Also, determinant and adjoint of square picture fuzzy matrix are established and some related properties are investigated. An application of picture fuzzy matrix in decision-making problem is presented here.
In this paper, we defined the Einstein Operations of intuitionistic fuzzy matrices and proved some results are equal using ⇁ implication operation.
Besides the various basic operations for intuitionistic fuzzy matrices already available in literature, two new operations are introduced in this paper. By using these operations scalar multiplication and power operations are constructed for an intuitionistic fuzzy matrix and some new relations are established. Also, the behavior of modal operators over these operations are studied.
In this article, the concepts of intuitionistic fuzzy complete and complete distributive lattice are introduced and the relative pseudocomplement relation of intuitionistic fuzzy sets is defined. The concepts of intuitionistic fuzzy eigenvalue and eigenvector of an intuitionistic fuzzy matrixes are presented and proved that the set of intuitionistic fuzzy eigenvectors of a given intuitionistic fuzzy eigenvalue form an intuitionistic fuzzy subspace. Also, the authors obtain an intuitionistic fuzzy maximum matrix of a given intuitionistic fuzzy eigenvalue and eigenvector and give some properties of an intuitionistic fuzzy maximum matrix. Finally, the invariant of an intuitionistic fuzzy matrix over a distributive lattice is given with some properties.
In this paper various properties of s-transitive intuitionistic fuzzy matrices are discussed. We obtain some results regarding convergence of powers of s-transitive intuitionistic fuzzy matrices. By using the properties of s-transitive intuitionistic fuzzy matrices we formulated and constructed canonical form.
In this paper, we study some properties of modal operators in intuitionistic fuzzy matrix and we introduce a new composition operator and discuss some of its algebraic properties. Finally, we obtain a decomposition of an intuitionistic fuzzy matrix by using the new composition operator and modal operators.
In this paper, we study some properties of intuitionistic fuzzy co-tautological matrix using intuitionistic fuzzy implication operator.
In this paper, the Cartesian product representation of an intuitionistic fuzzy matrix is introduced, in terms of its membership and non-membership parts. By using this representation, some basic properties of an intuitionistic fuzzy matrix (IFM) are discussed.
Like fuzzy matrix the interval-valued fuzzy matrix has also three types of ranks, viz, row rank, column rank and fuzzy rank. In this article, some methods are described to find these three type of ranks for interval-valued fuzzy matrices and investigated the relationship between them. Using the cross vector such ranks are investigated by very simple way. Many results are presented by using the definition of scalar multiplication of an interval-valued fuzzy matrix. © 2015, African Mathematical Union and Springer-Verlag Berlin Heidelberg.
In this paper, generalized intuitionistic fuzzy matrices (GIFMs) are defined. In fact, all GIFMs are intuitionistic fuzzy matrices (IFMs) but all IFMs are not GIFMs. Also, some operations are valid for IFMs but they are not valid for GIFMs. The relational properties, i.e., four types of reflexivity and irreflexivity, symmetricity and transitivity are studied here. Finally, some new operations over GIFMs are studied here.
In this paper, pseudo-similar intuitionistic fuzzy matrix (IFM) is defined with some properties of pseudo-similar and semi-similar IFMs studied. Here it is shown that, the pseudo-similarity relation preserve idempotency and regularity for the IFMs. After characterizations of symmetric IFM are discussed, it is shown that symmetric and idempotent IFMs possess symmetric Moore–Penrose inverse. Some results are investigated regarding the group inverse of IFMs.
In this paper, some elementary operations on triangular fuzzy numbers (TFNs) are defined. We also define some operations on triangu-lar fuzzy matrices (TFMs) such as trace and triangular fuzzy determinant (TFD). Using elementary operations, some important properties of TFMs are presented. The concept of adjoints on TFM is discussed and some of their properties are. Some special types of TFMs (e.g. pure and fuzzy triangular, symmetric, pure and fuzzy skew-symmetric, singular, semi-singular, constant) are defined and a number of properties of these TFMs are presented.
Intuitionistic fuzzy relations on finite universes can be represent by
intuitionistic fuzzy matrices and the
limiting behavior of the power matrices depends on the algebraic operation
employed on the matrices. In this
paper, the power of intuitionistic fuzzy matrices with maxgeneralized
mean-mingeneralized mean operation have
been studied. Here it is shown that the power of intuitionistic fuzzy
matrices with the said operations are
always convergent. The convergence of powers for an intuitionistic fuzzy
matrix with convex combination of
max-min and maxarithmetic mean-minarithmetic mean are also dicussed here.
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