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Fuzzy Inf. Eng. (2012) 4: 371-387 DOI 10.1007/s12543-012-0121-1 ORIGINAL ARTICLE Some Properties of Generalized Intuitionistic Fuzzy Nilpotent Matrices over Distributive Lattice Amal Kumar Adak· Monoranjan Bhowmik· Madhumangal Pal Received: 4 May 2012/ Revised: 12 September 2012/ Accepted: 27 October 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, the concept of lattice over generalized intuitionistic fuzzy matrices (GIFMs) are introduced and have shown that the set of GIFMs forms a dis- tributive lattice. Some algebraic properties of generalized intuitionistic fuzzy ma- trices (GIFMs) are presented over distributive lattice. Also, some characteristics of generalized intuitionistic fuzzy nilpotent matrices (GIFNMs) are discussed over dis- tributive lattice. Finally, the reduction of GIFNMs over distributive lattice are given with some properties. Keywords Intuitionistic fuzzy matrices· Generalized intuitionistic fuzzy matrices· Distributive lattice · Generalized intuitionistic fuzzy nilpotent matrices 1. Introduction The theory of fuzzy sets is applied to many mathematical branches. Many researchers have done several works on fuzzy sets. Atanassov [5, 6] introduced the concept of intuitionistic fuzzy sets (IFSs). Also a lot of research works were done by several re- searchers on the ﬁeld of IFS. Ragab and Emam [18] deﬁned adjoint of a square fuzzy matrix. By the concept of IFSs, ﬁrst time Pal [16] introduced intuitionistic fuzzy determinant. Later on Pal and Shyamal [20, 21] introduced intuitionistic fuzzy ma- trices and determined distance between intuitionistic fuzzy matrices. Bhowmik and Pal [7, 8] introduced some results on intuitionistic fuzzy matrices and intuitionistic circulant fuzzy matrices and generalized intuitionistic fuzzy matrices. Mondal and Amal Kumar Adak () · Madhumangal Pal () Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar Univer- sity, Midnapore-721102, India email: [email protected] [email protected] Monoranjan Bhowmik () Department of Mathematics, V. T. T College, Midnapore, Paschim Medinipur-721101, India email: [email protected] 372 Amal Kumar Adak · Monoranjan Bhowmik· Madhumangal Pal (2012) Samanta [13] introduced another concept of IFSs called generalized IFSs. Bhowmik and Pal [9] deﬁned generalized interval-valued intuitionistic fuzzy set (GIVIFS) and presented its various properties. Algebraic structures play a prominent role in the mathematics with wide range of applications in many disciplines such as theoretical physics, computer science, con- trol engineering, information sciences, coding theory, topological spaces etc. This provides suﬃcient motivation to the researchers to review various concepts and re- sults from the area of abstract algebra in the broader framework of fuzzy setting. One of the structures which is most extensively used and discussed in the mathematics and its applications is lattice theory. As it is well known that lattice is considered as a relational, ordered structure and as an algebra. Lattice matrices are useful tools in various domains like the theory of switching, automata theory and theory of ﬁnite graphs. The notions of nilpotent lattice matrices seem to appear ﬁrst in the work of Give’on [11]. In [11], Give’on proved that an n× n lattice matrix is nilpotent if and only if A = 0. Since then, a number of researchers have studied the topic of the nilpotent lattice matrices. Our aim is to introduce and study distributive lattice over GIFMs. The structure of this paper is organized as follows. In Section 2, the preliminaries and some deﬁ- nitions are given. In Section 3, some algebraic structures of GIFMs over distributive lattice are supplied and some results are given. In Section 4, we present some proper- ties of generalized intuitionistic fuzzy determinant over distributive lattice (GIFD). In Section 5, the deﬁnition of generalized intuitionistic fuzzy nilpotent matrix (GIFNM) over distributive lattice is given. In Section 6, the reduction of generalized intuition- istic fuzzy nilpotent matrices over distributive lattice are given and some properties are studied. The conclusion is made in Section 7. 2. Preliminaries Here some preliminaries, deﬁnitions of IFSs and GIFMs are recalled and some alge- braic operations of GIFMs and diﬀerent types of GIFMs are presented. 2.1. Fuzzy Set and Intuitionistic Fuzzy Set Deﬁnition 2.1 (Fuzzy set) A fuzzy set A in a universal set X is deﬁned as A = x,μ (x)|x ∈ X , where μ : X → [0, 1] is a mapping called the membership A A function of the fuzzy set A. Deﬁnition 2.2 (Instuitionistic fuzzy set) An instuitionistic fuzzy set (IFS) A over X is an object having the form A = x,μ (x),ν (x) : x ∈ X , where μ : X → [0, 1] A A A andν : X → [0, 1],μ (x) andν (x) are called the membership and non-membership A A A values of x in A satisfying the condition 0 ≤ μ (x)+ν (x) ≤ 1. A A Some operations on IFSs In the following, we deﬁne some relational operations on IFSs. Let A and B be two IFSs on X, where A = x,μ (x),ν (x) : x ∈ X A A and B = x,μ (x),ν (x) : x ∈ X . B B Fuzzy Inf. Eng. (2012) 4: 371-387 373 Then, (1) A = B ⇔ μ (x) = μ (x) and ν (x) = ν (x) for all x ∈ X. A B A B (2) A ⊆ B iﬀ μ (x) ≤ μ (x) andν (x) ≥ ν (x) for all x ∈ X. A B A B (3) A = x,ν (x),μ (x) : x ∈ X . A A (4) A∩ B = min{μ (x),μ (x)}, max{ν (x),ν (x)} : x ∈ X . A B A B (5) A∪ B = max{μ (x),μ (x)}, min{ν (x),ν (x)} : x ∈ X . A B A B 2.2. Fuzzy Matrix and Intuitionistic Fuzzy Matrix Deﬁnition 2.3 (Fuzzy matrix) A fuzzy matrix of order m× n is deﬁned as A = a , ijμ where a is the membership value of the i j-th element in A. ijμ Deﬁnition 2.4 (Intuitionistic fuzzy matrix) An intuitionistic fuzzy matrix of order m× n is deﬁned as A = a , a , where a and a are the membership and non- ijμ ijν ijμ ijν membership values of the i j-th element in A satisfying the condition 0 ≤ a +a ≤ 1 ijμ ijν for all i, j. Deﬁnition 2.5 (Generalized intuitionistic fuzzy matrix) A generalized intuitionistic fuzzy matrix (GIFM) of order m× n is deﬁned as A = a , a , where a and a ijμ ijν ijμ ijν are the membership and non-membership values of the i j-th element in A satisfying the generalized intuitionistic fuzzy condition 0 ≤ a ∧ a ≤ 0.5 for all i, j. ijμ ijν Let G denotes the set of all GIFMs of order m× n. In particular, G denotes the m×n n set of all GIFMs of order n× n. Deﬁnition 2.6 (Comparable GIFMs) Let A and B be two GIFMs such that A = a , a and B = b , b ∈ G . Then two matrices A and B are said to be ijμ ijν ijμ ijν m×n comparable GIFMs if a ≤ b and a ≥ b for all i, j. ijμ ijμ ijν ijν Some algebraic operations of GIFMs Let A and B be two GIFMs, such that A = a , a and B = b , b ∈ G . ijμ ijν ijμ ijν m×n (1) Matrix addition and subtraction are given by A+ B = max{a , b }, min{a , b } ijμ ijμ ijν ijν and A− B = a − b , a − b , ijμ ijμ ijν ijν ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ a , a ≥ b a , a < b , ⎨ ijμ ijμ ijμ ⎨ ijν ijν ijν where a − b = and a − b = ijμ ijμ ⎪ ijν ijν ⎪ ⎪ ⎪ ⎩ ⎩ 0, elsewhere 0, otherwise. (2) Componentwise matrix multiplication is given by A B = min{a , b }, max{a , b } . ijμ ijμ ijν ijν (3) Let A, B be two GIFMs of order m× n and n× p. Then the matrix product AB is given by AB = min{a , b }, max{a , b } ∈ G . ikμ kjμ ikν kjν m×p k k 374 Amal Kumar Adak · Monoranjan Bhowmik· Madhumangal Pal (2012) Diﬀerent types of IFMs (1) Intuitionistic fuzzy zero matrix is denoted by O and all entries of it are0, 1. (2) Intuitionistic fuzzy identity matrix I is deﬁned by a , a such that a = n ijμ ijν ijμ 1, a = 0 for i = j and a = 0, a = 1 for all i j. ijν ijμ ijν (3) If all element of an IFM are 1, 0, then it called intuitionistic fuzzy universal matrix and is denoted by J . (4) An IFM A is reﬂexive if and only if a = 1, 0 for all i. ii (5) If a = 0, 1 for all i of an IFM A, then it is called irreﬂexive. ii 2.3. Poset of Fuzzy Sets and GIFMs Deﬁnition 2.7 A binary relation ‘’ deﬁned on a fuzzy set A is a partial order on the fuzzy set A if the following conditions hold identically in A: (i) a a, (ii) a b and b a imply a = b, (iii) a b and b c imply a c. A nonempty fuzzy set A with a partial order on it is called a partially ordered set or brieﬂy a poset and it is denoted by (A,). Lemma 1 (Poset of GIFMs) Let G be the set of all n× n GIFMs and ‘≤’ be compa- rable fuzzy matrix relation. Then (G ,≤) is a poset. Proof Let A, B and C ∈ G . Then (1) A ≤ A is true since a ≤ a and a ≥ a . Hence the relation ‘≤’ is reﬂexive. ijμ ijμ ijν ijν (2) A ≤ B and B ≤ A possible only when A = B, since A ≤ B when a ≤ b and ijμ ijμ a ≥ b ; B ≤ A when b ≤ a and b ≥ a . Combining these two give ijν ijν ijμ ijμ ijν ijν A = B. Therefore the relation ‘≤’ is anti-symmetric. (3) A ≤ B then a ≤ b and a ≥ b ; B ≤ C then b ≤ c and b ≥ c .It ijμ ijμ ijν ijν ijμ ijμ ijν ijν is obvious that A ≤ C since a ≤ c and a ≥ c . Hence the relation ‘≤’is ijμ ijμ ijν ijν transitive. Therefore a nonempty set of GIFMs G satisﬁes the partial order relation. Hence G is a partial order set i.e. poset. Linearly ordered set of matrix. If every pair of the elements of a poset (G ,≤) are comparable, then G is said to be n n linearly ordered set of matrix. Predecessor and successor Let (G ,≤) be a poset and A, B ∈ G .If A ≤ B, then A is called predecessor and B is n n called successor. Fuzzy Inf. Eng. (2012) 4: 371-387 375 Maximal and minimal elements A matrix A ∈ G is said to be maximal matrix if there exists no matrix B such that A ≤ B. Similarly, a matrix A ∈ G is said to be minimal matrix if there exists no matrix B such that B ≤ A. Theorem 2.1 Every ﬁnite nonempty poset (G ,≤) has at least one maximal and one minimal elements. Proof Let G = {A , A ,··· , A } be a ﬁnite poset under ≤, containing n GIFMs. If n 1 2 n A is not a maximal GIFM, then by the deﬁnition there exists another GIFM A ∈ G 1 2 n such that A ≤ A . Again, if A is not a maximal GIFM, then there exists another 1 2 2 GIFM A ∈ G such that A ≤ A . Since G is ﬁnite, this process will terminate after 3 n 2 3 n a ﬁnite number of times. Hence, we obtain a ﬁnite sequence of GIFMs in G in the following ordered A ≤ A ≤ A ≤···≤ A . Therefore, there is no GIFM B such that 1 2 3 n A ≤ B for any B ∈ G . Hence A is a maximal GIFM of the poset (G ,≤). n n n n Similarly, it can be proved that poset (G ,≤) has minimal element. 2.4. Lattice of Fuzzy Sets Deﬁnition 2.8 (Lattice of fuzzy sets) A lattice is a partial ordered set (L,) in which every two elements have a unique least upper bound and a greatest lower bound. For any two elements a and b in L, the least upper bound and greatest lower bound will be denoted by a∨ b and a∧ b. Lattice is also denoted by (L,,∧,∨). Deﬁnition 2.9 (Universal bounds) An element a in the lattice L is called the universal upper bound if x a for all x ∈ L and an element b ∈ L is called universal lower bound if b x for all x ∈ L. The elements 0 and 1 are used to denote the universal lower and upper bounds respectively. Deﬁnition 2.10 (Distributive lattice of FSs) A lattice (L,,∨,∧) is said to be dis- tributive lattice if the operations ∨ and ∧ are distributive with respect to each other, i.e., (1) a∨ (b∧ c) = (a∨ b)∧ (a∨ c), (2) a∧ (b∨ c) = (a∧ b)∨ (a∧ c), where a, b and c ∈ L. An important special case of a distributive lattice is the real unit interval [0, 1] with ‘max’ and ‘min’ is called fuzzy algebra. 3. Distributive Lattice of GIFMs In this section, we introduce the concept of distributive lattice of GIMFs and give some properties of GIFNMs over distributive lattice. We begin this section with some deﬁnitions: 3.1. Lattice of GIFMs A nonempty poset (G ,≤) with two binary operation + and is called a lattice if the following axioms hold: 376 Amal Kumar Adak · Monoranjan Bhowmik· Madhumangal Pal (2012) (1) Closure : A, B ∈ G then A+ B ∈ G and A B ∈ G . n n n (2) Commutative : A, B ∈ G then A+ B = B+ A and A B = B A. (3) Associative : A, B, C ∈ G then (A+ B)+ C = A+ (B+ C) and (A B) C = A (B C). (4) Absorption : A, B ∈ G then A (A+ B) = A and A+ (A B) = A. Therefore, the poset (G ,≤) with two binary operation matrix addition and com- ponentwise matrix multiplication of GIFMs form lattice. It should be noted that the poset (G ,≤) with two binary operation matrix addition and matrix product of GIFMs does not form lattice as matrix product is not commu- tative. Idempotent law Let A be an n × n GIFMs over distributive lattice (G (L),≤,+, ). Then A satisﬁes idempotent law, i.e., (i) A+ A = A and (ii) A A = A. Theorem 3.1 Let A, B be two square GIFMs of n×n over distributive lattice (G (L),≤ ,+, ). Then A B = A if and only if A+ B = B. Proof Let A B = A, where A, B ∈ G (L). Therefore, min{a , b } = a and n ijμ ijμ ijμ max{a , b } = a . ijν ijν ijν Hence, max{a , b } = b and min{a , b } = b . ijμ ijμ ijμ ijν ijν ijν Now, A+ B= max{a , b }, min{a , b } ijμ ijμ ijν ijν = b , b = B. ijμ ijν The proof of converse part is similar. Theorem 3.2 Let (G (L),≤,+, ) be the lattice of GIFMs and A, B, C ∈ G .If A ≤ B n n and A ≤ C, then (1) A ≤ B+ C, (2) A ≤ B C. Proof If A ≤ B, then we have a ≤ b and a ≥ b . ijμ ijμ ijν ijν Again, A ≤ C we have a ≤ c and a ≥ c . ijμ ijμ ijν ijν Hence, a ≤ max{b , c } and a ≥ min{b , c }. ijμ ijμ ijμ ijν ijν ijν Therefore, A= a , a ijμ ijν ≤ max{b , c }, min{b , c } ijμ ijμ ijν ijν = B+ C. The proof of second part is similar. Theorem 3.3 Let (G (L),≤,+, ) be a lattice over GIFMs and A, B, C, D ∈ G .If n n A ≤ B and C ≤ D, then (1) A+ C ≤ B+ D and (2) A C ≤ B D. Proof If A ≤ B, then we have a ≤ b and a ≥ b . ijμ ijμ ijν ijν Fuzzy Inf. Eng. (2012) 4: 371-387 377 Again, C ≤ D,wehave c ≤ d and c ≥ d . ijμ ijμ ijν ijν Hence, max{a , c }≤ max{b , d } and min{a , c }≥ min{b , d }. ijμ ijμ ijμ ijμ ijν ijν ijν ijν Therefore, A+ C = max{a , c }, min{a , c } ijμ ijμ ijν ijν ≤ max{b , d }, min{b , d } ijμ ijμ ijν ijν = B+ D. Proof is similar for A C ≤ B D. 3.2. Distributive Lattice of GIFMs Let A, B, C ∈ G . Then the lattice of GIFMs (G (L),≤,+, ) is said to be distributive n n lattice of GIFMs if (1) A (B+ C) = (A B)+ (A C). (2) A+ (B C) = (A+ B) (A+ C). Example 1 We shown by means of example of the distributive property of GIFMs. Let A, B, C be three 3× 3 GIFMs, where ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ 0.3, 0.80.4, 0.80.5, 0.9 0.4, 0.70.5, 0.70.5, 0.8 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A = 0.4, 0.60.5, 0.70.3, 0.8 , B = 0.5, 0.50.6, 0.40.4, 0.7 , ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 0.3, 0.70.4, 0.50.3, 0.6 0.4, 0.70.5, 0.50.4, 0.6 ⎛ ⎞ ⎜ 0.6, 0.50.6, 0.40.7, 0.5⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = 0.6, 0.30.7, 0.40.5, 0.5 . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0.5, 0.60.6, 0.50.5, 0.3 Now, ⎛ ⎞ ⎜ ⎟ 0.6, 0.50.6, 0.40.7, 0.5 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ B+ C = 0.6, 0.30.7, 0.40.5, 0.5 , and ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0.5, 0.60.6, 0.50.5, 0.3 ⎛ ⎞ ⎜ ⎟ 0.3, 0.80.4, 0.80.5, 0.9 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A (B+ C) = 0.4, 0.60.5, 0.70.3, 0.8 , ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0.3, 0.70.4, 0.50.3, 0.6 ⎛ ⎞ ⎛ ⎞ ⎜ 0.3, 0.80.4, 0.80.5, 0.9⎟ ⎜ 0.3, 0.80.4, 0.80.5, 0.9⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A B = 0.4, 0.60.5, 0.70.3, 0.8 , A C = 0.4, 0.60.5, 0.70.3, 0.8 , ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 0.3, 0.70.4, 0.50.3, 0.6 0.3, 0.70.4, 0.50.3, 0.6 ⎛ ⎞ ⎜ ⎟ 0.3, 0.80.4, 0.80.5, 0.9 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A B+ A C = ⎜ ⎟ . 0.4, 0.60.5, 0.70.3, 0.8 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0.3, 0.70.4, 0.50.3, 0.6 Therefore, A (B+ C) = (A B)+ (A C). Similarly, it can be shown that A+ (B C) = (A+ B) (A+ C). Theorem 3.4 In a distributive lattice of GIFMs (G (L),≤,+, ) if A, B, C ∈ G (L), n n A+ B = A+ C and A B = A C, then B = C. 378 Amal Kumar Adak · Monoranjan Bhowmik· Madhumangal Pal (2012) Proof Since, A, B, C ∈ (G (L),≤,+, ), we have B = min{b , max{a , b }}, max{b , min{a , b }} [By absorption property] ijμ ijμ ijμ ijν ijν ijν = B max{a , c }, min{a , c } [Since A+ B = A+ C] ijμ ijμ ijν ijν = min{b , a }, max{b , a } + max{b , c }, min{b , c } ijμ ijμ ijν ijν ijμ ijμ ijν ijν [By distributive law] = max{c , a }, min{c , a } + min{b , c }, max{b , c } ijμ ijμ ijν ijν ijμ ijμ ijν ijν [SinceB A = C A] = min{c , a }, max{c , a } + min{c , b }, max{c , b } ijμ ijμ ijν ijν ijμ ijμ ijν ijν [By commutative law] = C max{a , b }, min{a , b } [By distributive law] ijμ ijμ ijν ijν = C (A+ C) = C [By absorption property]. Therefore, B = C. 4. Generalized Intuitionistic Fuzzy Determinant (GIFD) over Distributive Lat- tice The generalized intuitionistic fuzzy determinant |A| of an n × n GIFM A over a dis- tributive lattice (G (L),≤,+, ) is deﬁned as follows: det A = |A| = a , a a , a ···a , a , 1σ(1)μ 1σ(1)ν 2σ(2)μ 2σ(2)ν nσ(n)μ nσ(n)ν σ∈S where S denotes the symmetric group of all permutations of the indices (1, 2,··· , n). Proposition 4.1 If a GIFM B over a distributive lattice (G (L),≤,+, ), is obtained from an n× n GIFM A by multiplying the i-th row of A (i-th column) by k = k , k 1 2 such that 0 ≤ k + k ≤ 1, then k|A| = |B|. 1 2 Proof By deﬁnition of GIFD, we have |B| = b , b b , b ···b , b 1σ(1)μ 1σ(1)ν 2σ(2)μ 2σ(2)ν nσ(n)μ nσ(n)ν σ∈S = a , a a , a ··· ka , a ···a , a 1σ(1)μ 1σ(1)ν 2σ(2)μ 2σ(2)ν iσ(i)μ iσ(i)ν nσ(n)μ nσ(n)ν σ∈S = k a , a a , a ···a , a ···a , a 1σ(1)μ 1σ(1)ν 2σ(2)μ 2σ(2)ν iσ(i)μ iσ(i)ν nσ(n)μ nσ(n)ν σ∈S = k|A|. Proposition 4.2 Let Abeann× n GIFM over a distributive lattice (G (L),≤,+, ). If all the elements of a row (column) are0, 1, then |A| = 0, 1. Proof Since each term in |A| contains a factor of each row (column) and hence contains a factor of0, 1 row (column), so that each term of|A| is equal to0, 1 and consequently|A| = 0, 1. Fuzzy Inf. Eng. (2012) 4: 371-387 379 Proposition 4.3 Let Abeann× n GIFM over a distributive lattice (G (L),≤,+, ). If A is triangular, then|A| = a , a . iiμ iiν i=1 Proof Let A be a GIFM in triangular form below, i.e., a , a = 0, 1 for i < j. ijμ ijν Now consider a term b of|A| b = a , a a , a ···a , a . 1σ(1)μ 1σ(1)ν 2σ(2)μ 2σ(2)ν nσ(n)μ nσ(n)ν σ∈S Let σ(1) 1, so that 1<σ(1) and therefore a , a = 0, 1 and b = 0, 1. 1σ(1)μ 1σ(1)ν This means that each term is 0, 1 if σ(1) 1. Now let σ(1) = 1but σ(2) 2. Then 2 <σ(2) and a , a = 0, 1 and b = 0, 1. This means that each 2σ(2)μ 2σ(2)ν term is 0, 1 if σ(1) 1 and σ(2) 2. In the similar manner, we can prove that each term for which σ(1) 1or σ(2) 2··· or σ(n) n must be 0, 1. Hence |A| = a , a . iiμ iiν i=1 4.1. Generalized Intuitionistic Fuzzy Principal Submatrix Let A ∈ (G (L),≤,+, ) and A(i , i ,··· , j | j , j ,··· , j ) denote (n − t) × (n − t) n 1 2 t 1 2 t submatrix obtained from A by eliminating rows i , i ,··· , i and columns j , j ,··· , j 1 2 t 1 2 t is called a principal submatrix of order n− t of A. The adjoint of an IFM over a distributive lattice is deﬁned as below. Deﬁnition 4.1 Adjoint of an n× n GIFM A over a distributive lattice (G (L),≤,+, ), is denoted as adjA and is deﬁned as follows adj A = |A |, ji where|A | is the determinant of the (n− 1)× (n− 1) GIFM formed by deleting row j ji and column i from A. Deﬁnition 4.2 Let A be GIFM and A ∈ (G (L),≤,+, ). Then det A = a A(i| j). n ij i=1 5. Generalized Intuitionistic Fuzzy Nilpotent Matrix (GIFNM) over a Distribu- tive Lattice If A ∈ (G (L),≤,+, ) and A = 0 for some m ≥ 1, then A is called GIFNM over the distributive lattice (G (L),≤,+, ). The least positive integer m satisfying A = 0 is called the nilpotent index of A and is denoted by h(A). Deﬁnition 5.1 Let A be a GIFM and A ∈ G (L). Then A is said to be GIFNM if and only if every principal minor of A is0, 1. Proposition 5.1 Let A, B, C, A ,A ,A ,··· , A ∈ (G (L),≤,+, ). Then 1 2 3 n n (1) A(B− C) ≥ AB− AC and (A− B)C ≥ AC − BC, (2) (A− B)− C ≥ A− (B+ C), (3) If A ≤ B, then A− C ≤ B− C and C − A ≥ C − B, (4) (A − A )+ (A − A )+···+ (A − A )+ A = A + A +···+ A . 1 2 2 3 l−1 l l 1 2 l 380 Amal Kumar Adak · Monoranjan Bhowmik· Madhumangal Pal (2012) Proof Since A, B, C ∈ (G (L),≤,+, ), we have n n (A(B− C)) = a d , a d ij ikμ kjμ ikν kjν k=1 k=1 [where d = b − c and d = b − c ] kjμ kjμ kjμ kjν kjν kjν n n = a (b − c ), a (b − c ) ikμ kjμ kjμ ikν kjν kjν k=1 k=1 n n ≥ (a b − a c ), (a b − a c ) ikμ kjμ ikμ kjμ ikν kjν ikν kjν k=1 k=1 n n n n ≥ (a b , a b ) − (a c , a c ) ikμ kjμ ikν kjν ikμ kjμ ikν kjν k=1 k=1 k=1 k=1 = AB− AC. Hence, A(B− C) ≥ AB− AC. Similarly, it can prove the second part of this proposition. Proofs of (2) and (3) are straight forward. Proof (4): Let A , A , A ,··· , A ∈ (G (L),≤,+, ). We have to prove 1 2 3 n n (A − A )+ (A − A )+···+ (A − A )+ A = A + A +···+ A . 1 2 2 3 l−1 l l 1 2 l We prove this proposition by means of induction on l. Now for l = 2, we have (A − A )+ A = A + A . 1 2 2 1 2 Let us assume that the relation holds for l− 1. Now, (A − A )+ (A − A )+···+ (A − A )+ A 1 2 2 3 l−1 l l = (A − A )+ ((A − A )+···+ (A − A )+ A ) 1 2 2 3 l−1 l l = ((A − A )+ A )+ A +···+ A [By induction hypothesis] 1 2 2 3 l = A + A +···+ A. 1 2 l Hence (A − A )+ (A − A )+···+ (A − A )+ A = A + A +···+ A . 1 2 2 3 l−1 l l 1 2 l Example 2 Let A, B, C be three GIFMs over distributive lattice (G (L),≤,+, ), where ⎡ ⎤ ⎡ ⎤ ⎢ 0.7, 0.40.7, 0.30.7, 0.5⎥ ⎢ 0.5, 0.60.6, 0.40.5, 0.6⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A = 0.6, 0.50.8, 0.50.7, 0.4 , B = 0.6, 0.50.6, 0.50.5, 0.5 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ 0.7, 0.50.8, 0.40.8, 0.3 0.5, 0.60.5, 0.50.6, 0.4 and ⎡ ⎤ ⎢ 0.3, 0.80.4, 0.70.4, 0.8⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ C = 0.4, 0.70.5, 0.70.5, 0.6 . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.4, 0.80.3, 0.80.3, 0.9 Now we calculate the following ⎡ ⎤ ⎢ ⎥ 0.7, 0.40.7, 0.30.7, 0.5 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A− B = 0.6, 0.00.8, 0.00.7, 0.4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.7, 0.50.8, 0.40.8, 0.3 and Fuzzy Inf. Eng. (2012) 4: 371-387 381 ⎡ ⎤ ⎢ ⎥ 0.7, 0.40.7, 0.30.7, 0.5 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (A− B)− C = 0.6, 0.00.8, 0.00.7, 0.4 . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.7, 0.50.8, 0.40.8, 0.3 Again ⎡ ⎤ ⎢ ⎥ 0.5, 0.60.6, 0.40.5, 0.6 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ B+ C = 0.6, 0.50.6, 0.50.5, 0.5 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.5, 0.60.5, 0.50.6, 0.4 and ⎡ ⎤ ⎢ ⎥ 0.7, 0.40.7, 0.30.7, 0.5 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A− (B+ C) = ⎥ . 0.6, 0.00.8, 0.00.7, 0.4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.7, 0.50.8, 0.40.8, 0.3 Therefore, (A− B)− C ≥ A− (B+ C). Example 3 Let A, B, C be three 3× 3 GIFMs, where ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ 0.3, 0.80.7, 0.50.4, 0.5 0.4, 0.60.7, 0.40.6, 0.4 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A = ⎜ ⎟ , B = ⎜ ⎟ 0.5, 0.60.5, 0.60.6, 0.4 0.5, 0.40.6, 0.50.7, 0.3 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 0.6, 0.40.4, 0.70.5, 0.8 0.7, 0.40.5, 0.60.6, 0.5 and ⎛ ⎞ ⎜ ⎟ 0.4, 0.50.8, 0.40.6, 0.5 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Here A ≤ B. C = 0.6, 0.40.6, 0.30.7, 0.3 . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0.8, 0.40.6, 0.50.7, 0.4 Now, ⎛ ⎞ ⎜ 0.0, 0.00.0, 0.00.0, 0.0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A− C = 0.0, 0.00.0, 0.00.0, 0.0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0.0, 0.00.0, 0.00.0, 0.0 and ⎛ ⎞ ⎜ 0.4, 0.00.4, 0.00.6, 0.4⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ B− C = 0.0, 0.00.6, 0.00.7, 0.0 . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0.0, 0.00.0, 0.00.0, 0.0 Therefore, if A ≤ B, then A− C ≤ B− C. Proposition 5.2 If A is an GIFNM over a distributive lattice (G (L),≤,+, ), then det A = 0, 1 but converse of the proposition is not true. Proof Let A be a GIFNM over a distributive lattice (G (L),≤,+, ). Then det A = a A(i| j), j = 1, 2,··· , n. ij i=1 Again, by the deﬁnition of the GIFNM, a GIFM A will be a nilpotent intuitionistic fuzzy matrix if and only if every principal minor of A is0, 1. Therefore det A = 0, 1. Example 4 To show the converse part of the proposition, consider a GIFM over a distributive lattice (G (L),≤,+, )as ⎡ ⎤ ⎢ ⎥ 0, 10, 10, 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A = ⎢ ⎥ . 0.4, 0.90.5, 0.80.4, 0.7 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.5, 0.60.4, 0.80.4, 0.8 382 Amal Kumar Adak · Monoranjan Bhowmik· Madhumangal Pal (2012) Therefore, by the Proposition 5.2, we have det A = 0, 1 although it is not a GIFNM. Proposition 5.3 Let A be a GIFM and A ∈ (G (L),≤,+, ). Then A is GIFNM if and (k) (k) only if a = 0, 1, where a is the diagonal elements of A for all k. ii ii Example 5 Let us consider a GIFM over a distributive lattice (G (L),≤,+, )as ⎛ ⎞ ⎜ 0, 10, 10.4, 0.7⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A = 0.5, 0.60, 10.3, 0.8 . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0, 10, 10, 1 (1) Here a = 0, 1, i = 1, 2, 3. We obtain ii ⎛ ⎞ ⎜ ⎟ 0, 10, 10, 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A = 0, 10, 10.4, 0.7 . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0, 10, 10, 1 (2) Here a = 0, 1, i = 1, 2, 3. and ii ⎛ ⎞ ⎜ ⎟ 0, 10, 10, 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 3 ⎜ ⎟ A = ⎜ ⎟ , 0, 10, 10, 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0, 10, 10, 1 (3) also a = 0, 1, i = 1, 2, 3. ii Thus GIFM A over a distributive lattice L is a GIFNM and index is h(A) = 3. Proposition 5.4 Let A be a GIFM over a distributive lattice L, i.e., A ∈ (G (L),≤ ,+, ). Then A is GIFNM if and only if A is irreﬂexive and transitive. Proof Let A be an IF irreﬂexive matrix, i.e., a = 0, 1 for all i. Since, A is IF ii (k) 2 k transitive matrix, we have A ≤ A and so A ≤ A for all k. Therefore, a ≤ a = ii ii 0, 1 for all i, k ∈ N. So, A is a GIFNM. Conversely, suppose that A is a GIFNM. If A is not a GIFNM, then A 0.If 2 2 n n A = A, then A = A = ··· = A and so A = A 0. Thus a contradiction to the assumption that A is GIFNM. 2 2 n n Again, if A ≥ A, then A ≤ A ≤···≤ A and therefore A ≤ A 0, which is also a contradiction. Hence, A must be a generalized intuitionistic fuzzy transitive matrix. Now suppose that A is not generalized intuitionistic fuzzy irreﬂexive matrix. Then a 0, 1 for some i ∈ N. Therefore A is not a GIFNM, which is also a contradiction. ii Hence A must be a generalized intuitionistic fuzzy irreﬂexive and transitive ma- trix. Proposition 5.5 Let A be a GIFM and A ∈ (G (L),≤,+, ). If A generalized intu- itionistic fuzzy nilpotent matrix, then (1) A (adj A) = 0 and (adj A) A = 0. (2) (adj A) = 0. Proof Let B = A(adj A). Then for any i, j ∈ N (set of natural numbers) with i j. We have b , b = a , a ···a , a ···a , a . ijμ ijν 1σ(1)μ 1σ(1)μ iσ(i)μ iσ(i)ν nσ(n) iσ(i)ν σ∈S n Fuzzy Inf. Eng. (2012) 4: 371-387 383 Let σ ∈ S be an arbitrary. l l Case (i) σ (i) j for l ≥ 1. Then there exists d such that 1 ≤ d ≤ n, σ (i) = i and i, l d−1 σ (i)···σ (i) are mutually diﬀerent and belong to N. Then a , a ···a , a ···a , a 1σ(1)μ 1σ(1)ν iσ(i)μ iσ(i)ν nσ(n)μ nσ(n)ν ≤a , a ···a 2 , a 2 ···a d−1 , a d−1 iσ(i)μ iσ(i)ν σ(i)σ (i)μ σ(i)σ (i)ν σ (i)iμ σ (i)iν d k k k ≤ (A ) [since for GIFNM a = 0, 1 for all i, k ∈ N where A = a ] ii ii ii = 0, 1 . Case (ii) There exists l such that σ (i) = j . Then there exists d such that 1 ≤ d ≤ n, l l d−1 σ (i) = j and i, σ ( j)···σ ( j) are mutually diﬀerent and belong to N. Then a , a ···a , a ···a , a ···a , a 1σ(1)μ 1σ(1)ν iσ(i)μ iσ(i)ν iσ( j)μ iσ( j)ν nσ(n)μ nσ(n)ν 2 2 d−1 d−1 ≤a , a ···a , a ···a , a iσ( j)μ iσ( j)ν σ( j)σ ( j)μ σ( j)σ ( j)ν σ ( j)iμ σ ( j)iν d k k k ≤ (A ) [since for GIFNM a = 0, 1 for all i, k ∈ N where A = a ] ii ii ii = 0, 1 . Therefore, for any i, j ∈ N with i j,wehave b , b = a , a ···a , a ···a , a = 0, 1. ijμ ijν 1σ(1)μ 1σ(1)μ iσ(i)μ iσ(i)ν nσ(n) iσ(i)ν σ∈s If i = j, it is clear that b = 0, 1. ii Thus B = A(adj A) = 0. ⎛ ⎞ ⎜ ⎟ 0, 10, 10.4, 0.7 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Example 6 Let A = ⎜ ⎟ be a GIFNM over distributive lattice 0.3, 0.60, 10.3, 0.8 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0, 10, 10, 1 (G (L),≤,+, ). Now, the adjoint matrix of GIFM A is ⎛ ⎞ ⎜ 0, 10, 10, 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ adj A = 0, 10, 10.3, 0.7 . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0, 10, 10, 1 Also ⎛ ⎞ ⎛ ⎞ ⎜ 0, 10, 10, 1 ⎟ ⎜ 0, 10, 10, 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (adj A) = 0, 00, 10.3, 0.7 0, 10, 10.3, 0.7 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 0, 10, 10, 1 0, 10, 10, 1 ⎛ ⎞ ⎜ 0, 10, 10, 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = 0, 10, 10, 0 = 0. ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0, 10, 10, 1 Proposition 5.6 Let A be a GIFNM over a distributive lattice L, i.e., A ∈ G (L). T T Then h(A) = 3 if and only if AA = 0 and for some i, j ∈ N, R ∧ R 0, 1, where R and R are the i-th and j-th rows of the GIFNM A. i j 384 Amal Kumar Adak · Monoranjan Bhowmik· Madhumangal Pal (2012) Proof Let the index of a GIFM A be 3, i.e., h(A) = 3. Since A 0, then there must exist some rows say s-th and t-th such that R ∧ R 0, 1. Now we show that T T AA = 0. Suppose AA 0. Then we can ﬁnd p, q such thata , a a , a > pqμ pqν qpμ qpν 0, 1. Thereforea , a a , a a , a > 0, 1, which is a term of (p, q)- pqμ pqν qpμ qpν pqμ pqν th entry of the matrix A .So a , a a , a a , a > a , a a , a a , a qi μ qi ν i i μ i i ν i pμ i pν qpμ qpν pqμ pqν qpμ qpν 1 1 1 2 1 2 2 2 1≤i , i ≤n 1 2 > 0, 1, which leads to a contradiction, since A = 0. T 2 Conversely, R ∧ R 0, 1 for some rows say i, j, then A 0. Therefore, from T 3 AA = 0,weget A = 0. Hence A is an IFNM of index 3. Note: Let A be a GIFNM over a distributive lattice L, i.e., A ∈ G (L), and AA = 0 and for all i, j ∈ N, R ∧ R = 0, 1. Then A is neither GIFNM nor converge to a GIFM. ⎛ ⎞ ⎜ ⎟ 0, 10, 10.5, 0.7 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Example 7 Let A = 0.3, 0.80, 10, 1 be a GIFM over a distributive ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0, 10.4, 0.60, 1 T T lattice (G (L),≤,+, ) of order 3×3 and AA = 0, and for all i, j ∈ N, R ∧R = 0, 1. n i Now, ⎛ ⎞ ⎛ ⎞ ⎜ 0, 10.4, 0.70, 1 ⎟ ⎜ 0.3, 0.80, 10, 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 2 3 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A = 0, 10, 10.3, 0.8 , A = 0, 10.3, 0.80, 1 , ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎝ ⎠ ⎝ ⎠ 0.3, 0.80, 10, 1 0, 10, 10.3, 0.8 ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ 0, 10, 10.3, 0.8 0, 10.3, 0.80, 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 4 ⎜ ⎟ 5 ⎜ ⎟ ⎜ ⎜ A = ⎟ , A = ⎟ , 0.3, 0.80, 10, 1 0, 10, 10.3, 0.8 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 0, 10.3, 0.80, 1 0.3, 0.80, 10, 1 ⎛ ⎞ ⎜ 0.3, 0.80, 10, 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A = 0, 10.3, 0.80, 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0, 10, 10.3, 0.8 k k+3 and if continue this process, we get A = A where k ∈ N, set of natural numbers and k ≥ 3. Here A is neither GIFNM nor converge to any GIFM. 6. Reduction of GIFNMs over a Distributive Lattice Let L be a distributive lattice and A ∈ (G (L),≤,+, ). Then GIFM A/A = A− A is (L),≤ called a reduction of GIFM A. It is clear that A/A ≤ A for all GIFM A in (G ,+, ). Hashimoto [12] discussed the reduction of irreﬂexive and transitive fuzzy matrices and obtained some properties and applied these properties to nilpotent fuzzy matrices. In this section, we shall consider the reduction of GIFNMs over distributive lat- tice. The results obtained in this section generalize the previous results on nilpotent matrices by Hasimoto [12]. + + Theorem 6.1 Let A be a GIFNM and A ∈ (G (L),≤,+, ). Then (A/A) = A , where 2 + 2 3 n A/A = A− A and A = A+ A + A +···+ A . Fuzzy Inf. Eng. (2012) 4: 371-387 385 T T Let S = A/A. We note that A and S are GIFNM. It is clear that (A/A) ≤ A since A/A = A− A ≤ A. T T In the following, we shall prove that A ≤ (A/A) . To do this, we shall prove that l l l−1 S ≤ A − A for all l. l l l−1 We shall prove S ≤ A − A by induction on l. It is clear that it holds for l = 1, 2 l l−1 since S = A − A , and we may assume that it holds for l − 1. Then S = SS ≥ l−1 l l−1 l S (A − A ) ≥ SA − SA. Since A− A = S ≤ A.We have l 2 l−1 l+1 S ≥ (A− A )A − A l l+1 l+1 ≥ (A − A )− A l l+1 l+1 ≥ A − (A + A ) l l+1 = A − A . Now, + + (A/A) = S 2 3 n = S + S + S +···+ S 2 2 3 n n+1 ≥ (A− A )+ (A − A )+···+ (A − A ) 2 2 3 n−1 n n n+1 = (A− A )+ (A − A )+···+ (A − A )+ A (sinceA = 0) 2 3 n = A+ A + A +···+ A ( by Proposition 5.1) = A . + + Therefore, (A/A) = A . Theorem 6.2 Let Abeann× n irreﬂexive and transitive matrix over (G (L),≤,+, ). Then (A/A) = A. Proof Since A is IF irreﬂexive and transitive matrix, we have A is IF nilpotent + + matrix and A = A , and so (A/A) = A. Theorem 6.3 Let Abeann× n irreﬂexive and transitive matrix over (G (L),≤,+, ). Then the following conditions are equivalent (1) A/A ≤ S ≤ A, (2) S = A. Proof Suppose that A/A ≤ S ≤ A, clearly by Theorem 6.2, S = A. Thus we have that (1) implies (2). + 2 + 2 2 3 Now suppose that S = A. Then we have that S ≤ A and A = (S ) = S + S + n 2n ···+ S +···+ S . Since A is IF irreﬂexive and transitive, A is GIFNM and so S = 0 for l ≥ n (because S ≤ A). 2 2 3 n−1 2 2 3 n−1 + Therefore, A = S +S +···+S and so S+ A = S+S +S +···+S = S = A. (2) (2) Thus,s , s +a , a = a , a for all i and j, i.e.,s , s = a , a − ijμ ijν ijμ ijν ijμ ijν ijμ ijν ijμ ijν (2) (2) a , a = (A/A) for all i and j. ij ijμ ijν Hence we have that A/A ≤ S ≤ A and that (2) implies (1). 7. Conclusion In this paper, we have proved that GIFMs form a lattice and also shown that this lattice is distributive. But we have not studied whether this lattice is modular or not. 386 Amal Kumar Adak · Monoranjan Bhowmik· Madhumangal Pal (2012) Some properties of GIFMs are investigated, including the nilpotency of it. It can be seen that all GIFMs are not nilpotent, but under certain condition some GIFMs are nilpotent. we expect some other conditions may exist for nilpotency of GIFMs. Convergence of GIFMs over lattice is a very important property for any type of fuzzy matrices. At present, we are investigating this property for GIFMs. Acknowledgments The authors are very grateful and would like to express their sincere thanks to the anonymous referees and Editor-in-Chief Bing-Yuan Cao for their valuable comments to improve the presentation of the paper. References 1. Adak A K, Bhowmik M (2011) Interval cut-set of interval-valued intuitionistic fuzzy sets. African Journal of Mathematics and Computer Sciences 4(4): 192-200 2. Adak A K, Bhowmik M, Pal M (2011) Semi-ring of interval-valued intuitionistic fuzzy matrices. Global Journal of Computer Application and Technology 1(3): 340-347 3. 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Fuzzy Information and Engineering – Taylor & Francis
Published: Dec 1, 2012
Keywords: Intuitionistic fuzzy matrices; Generalized intuitionistic fuzzy matrices; Distributive lattice; Generalized intuitionistic fuzzy nilpotent matrices
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