Fuzzy Neutrosophic Soft Matrices of Type I and Type II

In this paper, two types of fuzzy neutrosophic soft matrix have been introduced and it is shown that the set of all fuzzy neutrosophic soft matrices form a semiring. Further it is explored that the set of all fuzzy neutrosophic soft matrices of both types form a vector space.


Introduction
The concept of fuzzy set was introduced by Zadeh [1] in 1965. The traditional fuzzy sets are characterised by the membership value or the grade of membership value. Sometimes it may be very difficult to assign the membership value for fuzzy sets. Consequently the concept of interval valued fuzzy sets was proposed [2] to capture the uncertainty of grade of membership value. Intuitionistic fuzzy sets introduced by Atanassov [3] is appropriate for such a situation. The intuitionistic fuzzy sets can only handle the incomplete information considering both the truth membership (or simply membership) and falsity-membership (or non membership) values. It does not handle the indeterminate and inconsistent information which exists in belief system. Smarandache [4] introduced the concept of neutrosophic set which is a mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data.
The concept of soft set theory was introduced by Molodtsov [5] in 1999, it is a new approach for modeling vagueness and uncertainty. Maji [6] et al. introduced the concept of fuzzy soft sets with the operations of union, intersection, complement of fuzzy soft sets. The fuzzy soft set concept extended soft sets into intuitionistic fuzzy soft set and fuzzy neutrosophic soft sets.
Rajarajeswari and Dhanalakshmi [7] introduced the intuitionistic fuzzy soft matrices applied in the application of medical diagnosis. Sumathi and Arockiarani [8] introduced new operations on fuzzy neutrosophic soft matrices.
In this paper, we have defined two types of Fuzzy Neutrosophic Soft Matrix (FNSM). In Section 2, we have recalled the definition of soft sets, neutrosophic sets, etc. Operations on neutrosophic soft set of both types have been defined, discussion about two types of CONTACT R. Uma uma83bala@gmail.com fuzzy neutrosophic soft algebra and two types of fuzzy neutrosophic soft matrix has been introduced in Section 3. In Section 4, we have shown that the set of all FNSM of type-I is a semiring under the operations componentwise addition ⊕, componentwise multiplication and the composition • . Similar results for FNSM of type II are discussed in Section 5.

Definition 1.1 ([4]):
A neutrosophic set A on the universe of discourse X is defined as In short an elementã in the neutrosophic set A, can be written asã = a T , a I , a F , where a T denotes degree of truth, a I denotes degree of indeterminacy, a F denotes degree of falsity such that 0 ≤ a T + a I + a F ≤ 3.

Definition 1.4 ([5]):
Let U be an initial universe set and E be a set of parameters. Let P(U) denotes the power set of U. Consider a nonempty set A, A ⊂ E. A pair (F,A) is called a soft set over U, where F is a mapping given by F : A → P(U).

Definition 1.5 ([9]):
Let U be an initial universe set and E be a set of parameters. Consider a nonempty set A, A ⊂ E. Let P(U) denotes the set of all fuzzy neutrosophic sets of U. The collection (F, A) is termed to be the Fuzzy Neutrosophic Soft Set (FNSS) over U, where F is a mapping given by F : A → P(U). Hereafter we simply consider A as FNSS over U instead of (F, A).

Fuzzy Neutrosophic Soft Algebra and Fuzzy Neutrosophic Soft Matrices
Definition 1.6: Letã = a T , a I , a F ,b = b T , b I , b F be any two elements in an FNSS A. We define the addition and multiplication in FNSS of type I as follows: Here a T ∨ b T means max{a T , b T } and a T ∧ b T means min{a T , b T } and so on. Clearly 0, 0, 1 is the additive identity and 1, 1, 0 is the multiplicative identity.
Similarly we define addition and multiplication in FNSS of type II as Clearly 0, 1, 1 is the additive identity and 1, 0, 0 is the multiplicative identity for type-II FNSS.
In this section, we discuss about two types of Fuzzy Neutrosophic Soft Algebra (FNSA).
An FNSA is a mathematical system (N,+,.) with two binary operations +,'.' defined on a set N satisfying the postulates.
By using neutrosophic soft set concept, we define the Fuzzy Neutrosophic Soft Matrix (FNSM).  E). The membership function, indeterminacy membership function and non membership function are written by

The matrix form of FNSM can be written as
are the membership value, indeterminacy value and non membership value respectively of u ∈ U for each e ∈ E. If we replace the identity element 0, 0, 1 by 0, 1, 1 in the above form, we get FNSM of type II.

Fuzzy Neutrosophic Soft Matrix of Type I
Let F m×n denote FNSM of order m × n and F n denote FNSM of order n × n.

Definition 2.1: Let
∈ F m×n the componentwise addition and componentwise multiplication are defined as

Definition 2.2:
Let A ∈ F m×n , B ∈ F n×p , the composition of A and B is defined as equivalently we can write the same as The product A • B is defined if and only if the number of columns of A is same as the number of rows of B. A and B are said to be conformable for multiplication. We shall use AB instead of A • B.

Definition 2.3:
The n × m Zero matrix O 1 is the matrix all of whose entries are of the form 0, 0, 1 . The n × n identity matrix I 1 is the matrix The n × m universal matrix J 1 is the matrix all of whose entries are of the form 1, 1, 0 . is the constant matrix all of whose entries are c further under componentwise multiplication

Theorem 2.6: The set F m×n is a fuzzy neutrosophic soft algebra under the componentwise addition and componentwise multiplication operations (⊕, ) defined as follows. For
Proof: The postulates (p 1 ) to (p 4 ) of a fuzzy neutrosophic soft algebra are automatically hold. A ⊕ O 1 = A and A J 1 = A forall A ∈ F m×n . Hence the zero matrix O 1 is the additive identity and the universal matrix J 1 is the multiplicative identity. Thus identity element relative to the operations + and exist. Further A ⊕ J 1 = J 1 and ij are all real numbers in [0, 1] they are comparable. If a ij ≤ b ij , (or) c ij then in both cases, (min{a ij , max{b ij , c ij }}) = a ij and max{inf {a ij , b ij }, min{a ij , c ij }} = a ij .
Therefore ij th entry of A (B ⊕ C) = (ij) th entry of (A B) ⊕ (A C). If a ij ≥ b ij and c ij then we have two cases, Thus the postulate (p 5 ) of distributivity holds. Thus F m×n is a fuzzy neutrosophic soft algebra with the operations (⊕, ) That is the left distributive property holds.
such that the ranges of suffixes i, j, k, are 1 to m, j = 1 to n, k = 1 to p respectively.
is the sum of the product of the corresponding elements in ith row of A and kth column of (B ⊕ C).
Similarly we can prove the right distributive property. In the following theorems, we prove that associative property holds under addition for FNSM of type I.

Theorem 2.8: For any three matrices A, B, C ∈
is the sum of the corresponding elements in the i th row of A and j th column of (B ⊕ C).
th element of (A ⊕ B) and C is the sum of the corresponding elements in i th row of (A ⊕ B) and j th column of C, from (3) and (4) In the following theorem, we prove that the associative property holds under multiplication for FNSM.

Theorem 2.9: For
Proof: With the given type of matrices (AB)C and A(BC) are both defined and are of type such that the ranges of the suffixes i, j, k, l are 1 to m, 1 to n, 1 to p and 1 to q respectively. Now (ik) th element of the product The (il)th element in the product (AB)C is the sum of product of the corresponding elements in the i th row of AB and l th column of C with k-common.
Thus (il) th element of Now (jl) th element of the product Again the (il) th element of the product of A(BC) is the sum of the product of the corresponding element in the (i) th row of A and (l) th column of BC.
If we write a member of V n , 1 × n matrix which is called row vector. The isometric set of n × 1 matrices is called column vector, and denoted by V n . For any result above V n there exist a corresponding results above V n . The system V n together with these operations called fuzzy neutrosophic soft vector space of type I.