Direct product of general intuitionistic fuzzy sets of subtraction algebras

We define direct product of -intuitionistic fuzzy sets and direct product of -intuitionistic fuzzy soft sets of subtraction algebras and investigate some related properties.


Introduction
The system (X;•, �) by Schein (1992), is a set of functions closed under the composition "•" under the composition of function(and hence (X, •) is a function semigroup) and the set theoretical subtraction "∖" (and hence (X, �) is a subtraction algebra in the sense of Abbot (1969)). He proved that every subtraction semigroup is isomorphic to a difference semigroup of invertible functions. Zelinka (1995) discussed a problem proposed by B. M. Schein concerning the structure of multiplication in a subtraction semigroup. He solved the problem for subtraction algebras of a special type called the atomic subtraction algebras. Jun, Kim, and Roh (2005) introduced the notion of ideals in subtraction algebras and discussed characterization of ideals. To study more about subtraction algebras see Ceven (2009), Jun and Kim (2007). The fuzzifications of ideals in subtraction algebras were discussed in Lee and Park (2007).

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Real world is featured with complex phenomenon. As vulnerability is unavoidably included in issues emerge in different fields of life and traditional techniques neglected to handle these sorts of issues. Managing with loose, unverifiable, or defective data was a major assignment for a long time. Numerous models were introduced with a specific end goal to appropriately join instability into framework portrayal; L.A. Zadeh in 1965 presented the thought of a fuzzy set. Zadeh supplanted traditional trademark capacity of established fresh sets which tackles its qualities in {0, 1} by enrollment capacity which tackles its values in shut interim [0,1]. Be that as it may, it is by all accounts the restricted case so this was summed up by K.T. Atanassov in 1986. Soft sets are additionally considered an exceptionally convenient device with a specific end goal to handle loose data. Here, we used a combination of soft and intutionistic fuzzy sets. Bhakat and Das (1996), introduced a new type of fuzzy subgroups, that is, the (∈, ∈ ∨q) fuzzy subgroups. In fact, the ∈, ∈ ∨q k fuzzy subgroup is an important generalization of Rosenfeld's fuzzy subgroup. Shabir et al. characterized semigroups by (∈, ∈ ∨q k )-fuzzy ideals in Shabir (2010). Gulistan, Shahzad, and Yaqoob (2014) studied ∈, ∈ ∨q k -fuzzy KU-ideals of KU-algebras. Molodtsov (1999) introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that are free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji, Roy, and Biswas (2002) described the application of soft set theory to a decision-making problem. Maji, Biswas, and Roy (2003) also studied several operations on the theory of soft sets. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh (1965).
The aim of this article is to study the concept of Direct product of ∈, ∈ ∨q k -intuitionistic fuzzy sets and Direct product of ∈, ∈ ∨q k -intuitionistic fuzzy soft sets of subtraction algebras and investigate some related properties.

Preliminaries
In this section we recall some of the basic concepts of subtraction algebra which will be very helpful in further study of the paper. Throughout the paper X denotes the subtraction algebra unless otherwise specified.
Definition 2.1 (Aygunoglu & Aygun, 2009) A nonempty set X together with a binary operation "-" is said to be a subtraction algebra if it satisfies the following: The last identity permits us to omit parentheses in expression of the form (x − y) − z. The subtraction determines an order relation on X : a ≤ b ⇔ a − b = 0, where 0 = a − a is an element that does not depend upon the choice of a ∈ X. The ordered set (X; ≤) is a semi-Boolean algebra in the sense of Abbot (1969), that is, it is a meet semi lattice with zero, in which every interval 0, a is a boolean algebra with respect to the induced order.
. In a subtraction algebra, the following are true (see Aygunoglu & Aygun, 2009): Proposition 2.4 (Aygunoglu & Aygun, 2009) Let X be a subtraction algebra and x, y ∈ X. If w ∈ X is an upper bound for x and y, then the element x ∨ y = w − ((w − y) − x) is the least upper bound for x and y.
Definition 2.7 (Lee & Park, 2007) A fuzzy set f is said to be a fuzzy ideal of X if it satisfies: Here we mentioned some of the related definitions and results which are directly used in our work. For details we refer the reader .
Definition 2.8 Atanassov (1986) An intuitionistic fuzzy set A in X is an object of the form where the function A :X → [0, 1] and A :X → [0, 1] denote the degree of membership and degree of non-membership of each element x ∈ X, and 0 ≤ A (x) + A (x) ≤ 1 for all x ∈ X. For simplicity, we will use the symbol A = ( A , A ) for the intuitionistic fuzzy set A = {(x, A (x), A (x)):x ∈ X}. We define 0(x) = 0 and 1(x) = 1 for all x ∈ X. Definition 2.9  Let X be a subtraction algebra. An intuitionistic fuzzy set ):x ∈ X}, of the form is said to be an intuitionistic fuzzy point with support x and value ( , ) and is denoted by x ( , ) . A intuitionistic fuzzy point x ( , ) is said to intuitionistic belongs to resp., intuitionistic quasi-coincident with intuitionistic fuzzy set resp., A (x) + < 1 and A (x) + < 1 . By the symbol x ( , ) q k A we mean A (x) + + k > 1 and A (x) + + k < 1, where k ∈ 0, 1 .
We use the symbol x t ∈ A implies A (x) ≥ t and t x [∈] A implies A (x) ≤ t, in the whole paper.
Definition 2.10  An intuitionistic fuzzy set A = ( A , A ) of X is said to be an ∈, ∈ ∨q k -intuitionistic fuzzy subalgebra of X if for all x, y ∈ X, t 1 , t 2 , t 3 , t 4 , k ∈ (0, 1).
Definition 2.11  An intuitionistic fuzzy set A = ( A , A ) of X is said to be an ∈, ∈ ∨q k -intuitionistic fuzzy ideal of X if it satisfies the following conditions, Molodtsov defined the notion of a soft set as follows.
Definition 2.12 (Molodtsov, 1999) A pair (F, A) is called a soft set over U, where F is a mapping given by F:A ⟶ P(U). In other words a soft set over U is a parametrized family of subsets of U.
The class of all intuitionistic fuzzy sets on X will be denoted by IF(X).
Definition 2.13 (Maji, Biswas, & Roy, 2001b,2004 Let U be an initial universe and E be the set of parameters. Let A ⊆ E. A pair (F, A) is called an intuitionistic fuzzy soft set over U, where F is a mapping given by F :A ⟶ IF(U).
In general, for every ∈ A.
is an intuitionistic fuzzy set in U and it is called intuitionistic fuzzy value set of parameter .
Definition 2.14  An intuitionistic fuzzy soft set ⟨F, A⟩ of X is called an ∈, ∈ ∨q k -intuitionistic fuzzy soft subalgebra of X, if for all ∈ A, Definition 2.15  An intuitionistic fuzzy soft set ⟨F, A⟩ of X is called an ∈, ∈ ∨q k -intuitionistic fuzzy soft ideal of X, if for all ∈ A, is an ∈, ∈ ∨q k -intuitionistic fuzzy soft ideal of X, if 3. Direct product of an ∈, ∈ ∨q k -intuitionistic fuzzy subalgebra/ideals In this section, we define Direct product of ∈, ∈ ∨q k -intuitionistic fuzzy sets and investigate some related properties.
) be two ∈, ∈ ∨q k -intuitionistic fuzzy sets of X 1 and X 2 , respectively. Then the Direct product of

Definition 3.4 An intuitionistic fuzzy set
Theorem 3.5 Let A and B be two ∈, ∈ ∨q k -intuitionistic fuzzy subalgebras of X 1 and X 2 , respectively. Then the Direct product A × B is an ∈, ∈ ∨q k -intuitionistic fuzzy subalgebra of X 1 × X 2 .
Proof Let A and B be two ∈, ∈ ∨q k -intuitionistic fuzzy subalgebras of X 1 and X 2 , respectively. For any (x 1 , y 1 ), (x 2 , y 2 ) ∈ X 1 × X 2 . We have Also Hence this shows that A × B is an ∈, ∈ ∨q k -intuitionistic fuzzy subalgebra of X 1 × X 2 . ✷ Theorem 3.6 Let A and B be two ∈, ∈ ∨q k -intuitionistic fuzzy ideals of X 1 and X 2 , respectively. Then the Direct product A × B is an ∈, ∈ ∨q k -intuitionistic fuzzy ideal of X 1 × X 2 .

Consider Also consider
Definition 3.9 Let A = ( A (x), A (x)) and B = ( B (x), B (x)) be intuitionistic fuzzy sets of X 1 and X 2 , respectively. Define the intuitionistic level set for the A × B as (A × B) ( , ) Theorem 3.10 Let A and B be two ∈, ∈ ∨q k -intuitionistic fuzzy subalgebras of X 1 and X 2 , respectively. Then the Direct product A × B is an ∈, ∈ ∨q k -intuitionistic fuzzy subalgebra of X 1 × X 2 if and only if Proof Straightforward. ✷ Theorem 3.11 Let A and B be two ∈, ∈ ∨q k -intuitionistic fuzzy ideals of X 1 and X 2 , respectively. Then the Direct product A × B is an ∈, ∈ ∨q k -intuitionistic fuzzy ideal of X 1 × X 2 if and only if ( A×B (x 1 , y 1 ) = A×B ((x 1 , y 1 ) − 0, 0 ), Proof Straightforward. ✷

Direct product of ∈, ∈ ∨q k -intuitionistic fuzzy soft subalgebras
In this section, we define Direct product of ∈, ∈ ∨q k -intuitionistic fuzzy soft sets and investigate some related properties.
Definition 4.1 Let ⟨F, A⟩ and ⟨G, B⟩ be two ∈, ∈ ∨q k -intuitionistic fuzzy soft sets of X 1 and X 2 , respectively. Then the Direct product of ∈, ∈ ∨q k -intuitionistic fuzzy soft sets ⟨F, A⟩ and ⟨G, B⟩ is defined as Here Definition 4.2 An ∈, ∈ ∨q k -intuitionistic fuzzy soft set ⟨ � F, A⟩ ⊗ ⟨ � G, B⟩ of X 1 × X 2 is called an an ∈, ∈ ∨q k -intuitionistic fuzzy soft subalgebra of X 1 × X 2 if it satisfies Theorem 4.4 Let ⟨F, A⟩ and ⟨G, B⟩ be two ∈, ∈ ∨q k -intuitionistic fuzzy soft subalgebras of X 1 and X 2 , respectively. Then the Direct product ⟨ � F, A⟩ ⊗ ⟨ � G, B⟩ is an ∈, ∈ ∨q k -intuitionistic fuzzy soft subalgebra of X 1 × X 2 .