( T,S ) – Intuitionistic fuzzy algebras

Abstract

In this study, a new concept of intuitionistic fuzzy abstract algebra using t-norms and t-conorms was introduced. Example that show the differences between intuitionistic fuzzy abstract algebra was given. Some properties were proved. Intuitionistic fuzzy congruence relation with respect to t-norms and t-conorms on (T, S)- intuitionistic fuzzy abstract algebra was examined.

Keywords

Intuitionistic fuzzy sets intuitionistic fuzzy abstract algebra intuitionistic fuzzy congruence relations t-norms and t-conorms

1 Introduction

The intuitionistic fuzzy set theory was introduced by Atanassov in 1983 as an extension of fuzzy sets by enlarging the truth value set to the lattice [0, 1] × [0, 1] given in Definition 1.1 [2, 3]. There are several theoretical and applied studies on intuitionistic fuzzy sets. Like algebra, analysis, decision making systems [15 , 30]. Fuzzy group definition has been redefined by Anthony and Sherwood using t-norms after being introduced by Rosenfeld [1, 24]. The approach to the algebraic structures with triangular norms allowed them to reach a strong form. The concept of the intuitionistic fuzzy group was given in 1989 and after then the intuitionistic fuzzy algebraic structures attracted the attention of many researchers. The generalization of these algebraic structures with triangular norms is being studied by different researchers up to date [5 , 13].

The generalization of abstract algebras into fuzzy set theory was introduced by Murali in 1987 [22]. The intuitionistic fuzzy algebra was defined by authors [10]. Intuitionistic fuzzy congruence relations on algebras and isomorphism theorems were studied and obtained main properties [12, 25].

In this study, the concepts of intuitionistic fuzzy algebra and intuitionistic fuzzy congruence relation are redefined with triangular norms. Specifically, when T = T_M and S = S_M are selected, the results in [10] can be easily obtained. However, for other norms, some differences have emerged except for the basic properties. For example, Proposition 4 is one of them.

Definition 1.1. Let L=[0,1] then L^∗ = {(x₁, x₂) ∈ [0, 1] ² : x₁ + x₂ ≤ 1} is a lattice with (x₁, x₂) ≤ (y₁, y₂) : ⇔ " x₁ ≤ y₁ and x₂ ≥ y₂ ".

For (x₁, y₁) , (x₂, y₂) ∈ L^∗, the operators ∧ and ∨ on (L^∗, ≤) are defined as follows;

(x₁, y₁) ∧ (x₂, y₂) = (min(x₁, x₂) , max(y₁, y₂))

(x₁, y₁) ∨ (x₂, y₂) = (max(x₁, x₂) , min(y₁, y₂))

For each J ⊆ L^∗

sup J = (sup {x : (x, y ∈ [0, 1]) , ((x, y) ∈ J)} , inf {y:( x, y ∈ [0, 1]) ((x, y) ∈ J)}) and

inf J = (inf {x : (x, y ∈ [0, 1]) ((x, y) ∈ J)} , sup {y : ( x, y ∈ [0, 1]) ((x, y) ∈ J)}) .

Definition 1.2. [2] An intuitionistic fuzzy set (shortly IFS) on a set X is an object of the form $\begin{matrix} A = {< x, μ_{A} (x), ν_{A} (x) > : x \in X} \end{matrix}$ where μ_A (x) , (μ_A : X → [0, 1]) is called the “degree of membership of x in A ”, ν_A (x) , (ν_A : X → [0, 1]) is called the ““degree of non- membership of x in A ”and where μ_A and ν_A satisfy the following condition: $\begin{matrix} μ_{A} (x) + ν_{A} (x) \leq 1, forallx \in X . \end{matrix}$

The hesitation degree of x is defined by π_A (x) = 1 - μ_A (x) - ν_A (x) .

Definition 1.3. [2] An IFS A is said to be contained in an IFS B (notation A ⊑ B) if and only if, for all x∈ X : μ_A (x) ≤ μ_B (x) and ν_A (x) ≥ ν_B (x) .

It is clear that A = B if and only if A ⊑ B and B ⊑ A .

Atanassov showed that, there are 53 negations now known in intuitionistic fuzzy logic [4]. The first negation defined is as follows:

Definition 1.4. [2] Let A ∈ IFS (X) and A = {< x, μ_A (x) , ν_A (x) > : x ∈ X} then the above set is called the complement of A $\begin{matrix} A^{c} = {< x, ν_{A} (x), μ_{A} (x) > : x \in X} \end{matrix}$

Definition 1.5. [2] Let A ∈ IFS (X) .Then (r, s)-cut and strong (r, s)-cut of A are crisp subsets A_(r,s) and A_{〈r, s〉} of the X, respectively are given by $\begin{array}{l} A_{(r, s)} = {x : x \in X s u c h t h a t μ_{A} (x) \geq r, v_{A} (x) \leq s} \\ A_{〈 r, s 〉} = {x : x \in X s u c h t h a t μ_{A} (x) > r, v_{A} (x) < s \end{array}$ where r, s ∈ [0, 1] with r + s ≤ 1 .

Intuitionistic fuzzy relations were introduced by Burille and Bustince.

Definition 1.6. [7] An intuitionistic fuzzy relation (shortly IFR) is an intuitionistic fuzzy subset of X × Y, that is, is an expression R given by R = {〈 (x, y) , μ_R (x, y) , ν_R (x, y) 〉 : x ∈ X, y ∈ Y} where μ_R : X × Y → [0, 1], ν_R : X × Y → [0, 1] satisfy the condition 0 ≤ μ_R (x, y) + ν_R (x, y) ≤1 for any (x, y) ∈ X × Y.

Definition 1.7. [8] Let X be a non-empty set and R ∈ IFR (X) .

For every x ∈ X, $\begin{matrix} μ_{R} (x, x) = 1 \end{matrix}$ $\begin{matrix} ν_{R} (x, x) = 0 \end{matrix}$ then R is called an intuitionistic fuzzy reflexsive.

For every x, y ∈ X, $\begin{matrix} μ_{R} (x, y) \leq μ_{R} (y, x) \end{matrix}$ $\begin{matrix} ν_{R} (x, y) \geq ν_{R} (y, x) \end{matrix}$ then R is called an intuitionistic fuzzy symmetric.

For every x, y, z ∈ X, $\begin{matrix} μ_{R} (x, y) \land μ_{R} (y, z) \leq μ_{R} (x, z) \end{matrix}$ $\begin{matrix} ν_{R} (x, y) \lor ν_{R} (y, z) \geq ν_{R} (x, z) \end{matrix}$ then R is called an intuitionistic fuzzy transitive.

If an intuitionistic fuzzy relation satisfies the previous properties then it is called an intuitionistic fuzzy equivalence relation (IFE (X)). Now we can talk about intuitionistic fuzzy equivalence classes of R .

Definition 1.8. [17] Let X be a non-empty set, R ∈ IFE (X) and a ∈ X . $\begin{matrix} {[a]}_{R} = {< x, μ_{{[a]}_{R}} (x), ν_{{[a]}_{R}} (x) > : x \in X} \end{matrix}$ where μ_{[a]_R} (x) = μ_R (a, x) , ν_{[a]_R} (x) = ν_R (a, x) is called an intuitionistic fuzzy equivalence class of a w.r.t R.

Theorem 1.1. [17] Let X be a non-empty set and R ∈ IFR (X) . Then R ∈ IFE (X) if and only if R_(r,s) is a equivalence relation on X for each r, s ∈ [0, 1] with r + s ≤ 1 .

Definition 1.9. [16] Let X and Y be two non-empty sets and f : X → Y be a mapping. Let A ∈ IFS (X) and B ∈ IFS (Y) . Then f is extended to a mapping from IFS (X) to IFS (Y) as

$\begin{matrix} f (A) (y) = (μ_{f (A)} (y), ν_{f (A)} (y)) \end{matrix}$ where $\begin{matrix} μ_{f (A)} (y) = {\begin{matrix} ⋁ {μ_{A} (x) : x \in f^{- 1} (y)} \\ 0; otherwise \end{matrix} \end{matrix}$ and $\begin{matrix} ν_{f (A)} (y) = {\begin{matrix} ⋀ {ν_{A} (x) : x \in f^{- 1} (y)} \\ 1; otherwise \end{matrix} \end{matrix}$ f (A) is called the image of A under the map f .Also, the pre-image of B under f is denoted by f^-1 (B) and defined as $\begin{matrix} f^{- 1} (B) (x) = (μ_{f^{- 1} (B)} (x), ν_{f^{- 1} (B)} (x)) \end{matrix}$ where $μ_{f^{- 1} (B)} (x) = μ_{B} (f (x)) a n d v_{f^{- 1} (B)} (x) = v_{B} (f (x))$

Murali was introduced fuzzy algebra using Zadeh’s extension principle. Fuzzy congruence relations were defined and some properties of fuzzy congruence relations were studied by same author [22 , 29].

The definition of abstract algebra in crisp set theory is as follows;

Definition 1.10. [6] An abstract algebra (or an algebra) A is a pair [S, F] where S is a non-empty set and F is a specified set of operatians f_α, each mapping a power S^n(α) of S into S, for some appropriate nonnegative finite integer n (α).

Otherwise stated, each operation f_α assigns to every n (α)-ple (x₁,. . . , x_n(α)) of elements of S, a value f_α (x₁,. . . , x_n(α)) in S, the result of performing the operation f_α on the sequence x₁,. . . , x_n(α) . If n (α) =1, the operation f_α is called unary; if n (α) =2, it is called binary; if n (α) =3, it is called ternary, etc. When n (α) =0, the operation f_α is called nullary; it selects a fixed element of S .

A = [S, F] and B = [T, F′] are called similar algebras if F and F′ for each α the types of f_α and $f_{α}^{'}$ are same.

Definition 1.11. [6] Let A = [S, F] and B = [T, F′] be two similar algebras. A function φ : S → T is called a homomorphism of A into B if and only if for all f_α ∈ F and x_i ∈ S,i = 1, 2,. . . , n (a), $\begin{matrix} f_{α}^{'} (φ (x_{1}), φ (x_{2}), . . ., φ (x_{n (a)})) = \\ φ (f_{α} (x_{1}, x_{2}, . . ., x_{n (a)})) . \end{matrix}$

A crisp congruence relation on an algebraic system A = [S, F] is an equivalence relation θ on A = [S, F] which has the substitution property for its operations. It means that, for all f_α ∈ F and a_i, b_i ∈ S,i = 1, 2,. . . , n (a) , $\begin{matrix} a_{i} & \equiv & b_{i} (θ) \Rightarrow \\ f_{α} (a_{1}, a_{2}, . . ., a_{n (a)}) & \equiv & f_{α} (b_{1}, b_{2}, . . ., b_{n (a)}) (θ) . \end{matrix}$

The extentions of fuzzy algebras to intuitionistic fuzzy sets defined in 2017.

Definition 1.12. [10] Let S = [X, F] be an algebra where X is a non-empty set and F is a specified set of finite operatians f_α, each mapping a power X^n(α) of X into X, for some appropriate nonnegative finite integer n (α). For each f_α, a corresponding operation ω_α on IFS (X) as follows; $\begin{matrix} ω_{α} : IFS (X) \times IFS (X) \times . . . \times IFS (X) \to IFS (X), \\ ω_{α} (A_{1}, A_{2}, . . ., A_{n (α)}) = A \end{matrix}$ such that $\begin{matrix} A (x) = {\begin{matrix} sup {A_{1} (x_{1}) \land . . . \land A_{n (α)} (x_{n (α)})}, & f_{α} (x_{1}, . . ., x_{n (α)}) = x \\ Θ = (0, 1), & otherwise \end{matrix} \end{matrix}$ Shortly, A = ω_α (A₁, A₂,. . . , A_n(α)) . Let Ω = {ω_α : corresponding operation for each f_α ∈ F} then L = [(I × I) ^X, Ω] is called intuitionistic fuzzy algebra.

If n (α) =0 then f_α (x) = e that e is a fixed element of X. So, ω_α is defined as following: $\begin{matrix} ω_{α} (A) = A_{e} \end{matrix}$ $\begin{matrix} A_{e} (x) = {\begin{matrix} sup_{x \in X} A (x), & x = e \\ (0, 1), & x \neq e \end{matrix} \end{matrix}$

Definition 1.13. [10] Let X be a non-empty set and A ∈ IFS (X) . A is called an intuitionistic fuzzy subalgebra (IF-subalgebra) of L = [IFS (X) , Ω] intuitionistic fuzzy algebra if and only if for nonnegative finite integer n (α) , ω_α (A, A,. . . , A) ⊑ A, for every ω_α .

Triangular norms concept was introduced with the paper "Statistical metrics" by Menger [21]. t-Norms have a major role on the theory of probabilistic metric spaces. Also, triangular norms were used as a broad studying area in fuzzy set theory.

Definition 1.14. [19] A t-norm is a mapping T : [0, 1] × [0, 1] → [0, 1] satisfying the following conditions for all x, y, z ∈ [0, 1]:

T (x, 1) = x

T (x, y) = T (y, x)

T (x, T (y, z)) = T (T (x, y) , z)

T (x, y) ≤ T (x, z) whenever y ≤ z .

Example 1. Here are the four basic t-norms:

T_M (x, y) = min(x, y) (minimum)

T_P (x, y) = xy (product)

T_L (x, y) = max(x + y - 1, 0) (Lukasiewicz t-norm)

$T_{D} (x, y) = {\begin{matrix} 0, & if (x, y) \in [0, 1)^{2} \\ min (x, y), & otherwise \end{matrix}$ (drastic product)

Definition 1.15. [19] For $n \in ℕ,$ the function $T_{n} : \prod_{i \in I_{n}} [0, 1] \to [0, 1]$ define by $\begin{matrix} T_{n} (x_{1}, . . ., x_{n}) = T (x_{i}, T_{n - 1} (x_{1}, . . ., x_{i - 1}, x_{i + 1}, . . . x_{n})) \\ = T_{i = 1}^{n} x_{i} \end{matrix}$ for i ∈ I_n where T₂ = T and T = id (identity) .

Definition 1.16. [19] A t-conorm is a mapping S : [0, 1] × [0, 1] → [0, 1] satisfying the following conditions for all x, y, z ∈ [0, 1]:

S (x, 0) = x

S (x, y) = S (y, x)

S (x, S (y, z)) = S (S (x, y) , z)

S (x, y) ≤ S (x, z) whenever y ≤ z .

Example 2. Here are the basic t-conorms:

S_M (x, y) = max(x, y) (maximum)

S_P (x, y) = x + y - xy (probabilistic sum)

S_L (x, y) = min(x + y, 1) (Lukasiewicz t-conorm)

$S_{D} (x, y) = {\begin{matrix} 1, & if (x, y) \in (0, 1]^{2} \\ max (x, y), & otherwise \end{matrix}$ (drastic product)

It is clear that, $\begin{matrix} T (x, y) \leq min (x, y) \leq max (x, y) \leq S (x, y) \end{matrix}$ for all x, y ∈ [0, 1] .

Definition 1.17. For a t-norm T and a t-conorm S on [0, 1] Δ_T and Δ_S are as follow; $\begin{matrix} Δ_{T} & = & {x \in [0, 1] : T (x, x) = x} \\ Δ_{S} & = & {x \in [0, 1] : S (x, x) = x} \end{matrix}$

Proposition 1.2. [19] A mapping S : [0, 1] × [0, 1] → [0, 1] is a t-conorm if and only if there exists a t-norm T such that for all (x, y) ∈ [0, 1] ² $\begin{matrix} S (x, y) = 1 - T (1 - x, 1 - y) . \end{matrix}$

The t-conorm given by Proposition 2 is called the dual t-conorm of T and the t-norm is said to be the dual t-norm of S.

Definition 1.18. [19] For $n \in ℕ,$ the function $S_{n} : \prod_{i \in I_{n}} [0, 1] \to [0, 1]$ define by $\begin{matrix} S_{n} (x_{1}, . . ., x_{n}) = S (x_{i}, S_{n - 1} (x_{1}, . . ., x_{i - 1}, x_{i + 1}, . . ., x_{n})) \\ = S_{i = 1}^{n} x_{i} \end{matrix}$ for i ∈ I_n where S₂ = S and S = id (identity) .

Remark 1.3. [19] If (T, S) is a pair of mutually dual t-norms and t-conorms, then the duality given by Proposition 2 can be generalized as follows: $\begin{matrix} S_{i = 1}^{n} x_{i} = 1 - T_{i = 1}^{n} (1 - x_{i}) \end{matrix}$

Definition 1.19. [18] Let A ∈ IFS (X) and T be a t-norm. Then A_T,r is a subset of X defined by $\begin{matrix} A_{T, r} = {x \in X : T (μ_{A} (x), 1 - ν_{A} (x)) \geq r} \end{matrix}$ for every r ∈ [0, 1] .

Definition 1.20. [18] Let A ∈ IFS (X) and (T, S) is a pair of mutually dual t-norms and t-conorms. Then A_T,S,r is a subset of X defined by $\begin{matrix} A_{T, S, r} = {x \in X : T (μ_{A} (x), S (μ_{A} (x), ν_{A} (x))) \geq r} \end{matrix}$ for every r ∈ [0, 1] .

2 (T, S)-Intuitionistic fuzzy algebras

Triangular norms are widely used in applications in multivalued logic or an intersection of fuzzy sets. The T = T_Mand S = S_M norms have been used in design of fuzzy logic controllers and also in the modelling of other decision making processes. Some theoretical and experimental studies seem to specify that triangular norms may work better then T = T_M and S = S_M norms in the decision making processes. For similar reasons, it is appropriate to work with triangular norms for more sensitive results. Based on this, in this section, intuitionistic fuzzy algebras were redefined using triangular norms. In addition to the properties provided by intuitionistic fuzzy algebras, different new properties were proven.

Definition 2.1. Let K = [X, F] be an algebra where X is a non-empty set and F is a specified set of finite operatians f_α, each mapping a power X^n(α) of X into X, for some appropriate nonnegative finite integer n (α). For each f_α, a corresponding operation $ω_{f_{α}}^{(T, S)}$ on IFS (X) as follows; $\begin{matrix} ω_{f_{α}}^{(T, S)} : IFS (X) \times . . . \times IFS (X) \to IFS (X), \\ ω_{f_{α}}^{(T, S)} (A_{1}, A_{2}, . . ., A_{n (α)}) = A, \end{matrix}$ $\begin{matrix} A (x) = {\begin{matrix} sup (T_{i = 1}^{n} μ_{A_{i}} (x_{i}), S_{i = 1}^{n} ν_{A_{i}} (x_{i})), \\ f_{α} (x_{1}, . . ., x_{n (α)}) = x \\ Θ = (0, 1), otherwise \end{matrix} \end{matrix}$ such that (T, S) is a pair of mutually dual t-norms and t-conorms. Let $Ω^{(T, S)} = {ω_{f_{α}}^{(T, S)} :$ corresponding operation for each f_α ∈ F} then L = [IFS (X) , Ω^(T,S)] is called (T, S)-intuitionistic fuzzy algebra.

If n (α) =0 then f_α (x) = e that e is a fixed element of X. So, $ω_{f_{α}}^{(T, S)}$ is defined as following: $\begin{matrix} ω_{f_{α}}^{(T, S)} (A) = A_{e} \end{matrix}$ $\begin{matrix} A_{e} (x) = {\begin{matrix} sup_{x \in X} A (x), & x = e \\ (0, 1), & x \neq e \end{matrix} \end{matrix}$

Definition 2.2. Let X be a non-empty set and A ∈ IFS (X) . A is called an (T, S)-intuitionistic fuzzy subalgebra of Ł = [IFS (X) , Ω^(T,S)] if and only if for nonnegative finite integer $n (α), ω_{f_{α}}^{(T, S)} (A, A, . . ., A) ⊑ A,$ for every $ω_{f_{α}}^{(T, S)} .$

Theorem 2.1. Let K = [X, F] be an algebra, f_α ∈ F and A, A₁, A₂,. . . , A_n(α) be (T, S)-intuitionistic fuzzy subalgebras. $\begin{matrix} ω_{f_{α}}^{(T, S)} (A_{1}, A_{2}, . . ., A_{n (α)}) ⊑ A \end{matrix}$ if and only if $\begin{matrix} A (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)})) \geq (T_{i = 1}^{n} μ_{A_{i}} (x_{i}), S_{i = 1}^{n} ν_{A_{i}} (x_{i})) \end{matrix}$ is true for every (x₁, x₂,. . . , x_n(α)) ∈ X^n(α) .

Proof. (1) Let n (α) ≠0 and $\begin{matrix} ω_{f_{α}}^{(T, S)} (A_{1}, A_{2}, . . ., A_{n (α)}) ⊑ A . \\ ω_{f_{α}}^{(T, S)} (A_{1}, A_{2}, . . ., A_{n (α)}) (x) \leq A (x), for all x \in X . \end{matrix}$ So, $\binom{sup}{f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}) = x} (T_{i = 1}^{n} μ_{A_{i}} (x_{i}), S_{i = 1}^{n} ν_{A_{i}} (x_{i})) \leq A (x),$ for all (x₁,. . . , x_n(α)) ∈ X^n(α) . $\begin{matrix} A (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)})) \geq \\ ω_{f_{α}}^{(T, S)} (A_{1}, A_{2}, . . ., A_{n (α)}) (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)})) \\ = \binom{sup}{f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}) = x} (T_{i = 1}^{n} μ_{A_{i}} (x_{i}), S_{i = 1}^{n} ν_{A_{i}} (x_{i})) \\ \geq (T_{i = 1}^{n} μ_{A_{i}} (x_{i}), S_{i = 1}^{n} ν_{A_{i}} (x_{i})) \end{matrix}$ Conversely, let $A (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)})) \geq (T_{i = 1}^{n} μ_{A_{i}} (x_{i}), S_{i = 1}^{n} ν_{A_{i}} (x_{i})) .$

For all f_α (x₁, x₂,. . . , x_n(α)) = x then $\begin{matrix} \binom{sup}{f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}) = x} (T_{i = 1}^{n} μ_{A_{i}} (x_{i}), S_{i = 1}^{n} ν_{A_{i}} (x_{i})) \\ \leq sup_{f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}) = x} ({T_{M}}_{i = 1}^{n} μ_{A_{i}} (x_{i}), {S_{M}}_{i = 1}^{n} ν_{A_{i}} (x_{i})) \\ \leq A (x) . \end{matrix}$ If for some x there exists no such n (a)-tuples then $ω_{f_{α}}^{(T, S)} (A_{1}, A_{2}, . . ., A_{n (α)}) (x) = (0, 1) \leq A (x) .$ (2)If n (α) =0 then f_α (x) = e, e is a fixed element of X . For any B (T, S)-intuitionistic fuzzy subalgebra, $\begin{matrix} ω_{f_{α}}^{(T, S)} (B) (x) \leq A (x) \leftrightarrow A (e) \geq ω_{f_{α}^{(T, S)}} (B) \\ = \underline{x} sup B (x) \Leftrightarrow A (ω_{f_{α}^{(T, S)}} (x)) \geq B (x) \end{matrix}$ for all x ∈ X .□

Example 3. Let consider $ℤ_{6}$ algebra which defined with two binary operations (+ , ·), two nullary operations $(\bar{0}, \bar{1})$ and a unary operation (-). The intuitionistic fuzzy set $A \in IFS (ℤ_{6})$ defined by $\begin{matrix} A = {(\bar{0}, 0.5, 0.1), (\bar{1}, 0.2, 0.5), (\bar{2}, 0.3, 0.3), \\ (\bar{3}, 0.4, 0.4), (\bar{4}, 0.3, 0.3), (\bar{5}, 0.2, 0.5)} \end{matrix}$ isn’t an intuitionistic fuzzy algebra but with T (x, y) = max(x + y - 1, 0) and S (x, y) = min(x + y, 1) dual norms it is a (T, S)-intuitionistic fuzzy algebra.

Theorem 2.2. Let K = [X, F] be an algebra. If {A_i } _i∈Λis a family of (T, S)- intuitionistic fuzzy subalgebras of on K then $\begin{matrix} A = \underset{i \in Λ}{\cap} A_{i} \end{matrix}$ is a (T, S)-intuitionistic fuzzy subalgebra.

Proof. Let f_α ∈ F and for the corresponding n (α) , (x₁, x₂,. . . , x_n(α)) ∈ X^n(α) . $\begin{matrix} A & (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)})) \\ = & ⋂_{i \in Λ} A_{i} (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)})) \\ \geq & (inf_{i \in I} T_{j = 1}^{n} μ_{A_{i}} (x_{j}), sup_{i \in I} S_{j = 1}^{n} ν_{A_{i}} (x_{j})) \\ \geq & (T_{j = 1}^{n} (inf_{i \in I} μ_{A_{i}} (x_{j})), S_{j = 1}^{n} (sup_{i \in I} ν_{A_{i}} (x_{j}))) \\ = & (T_{j = 1}^{n} μ_{A} (x_{j}), S_{j = 1}^{n} ν_{A} (x_{j})) \end{matrix}$ So, A a (T, S)-intuitionistic fuzzy subalgebra on K . □

Theorem 2.3. Let K = [X, F], L = [Y, F] be two similar algebras and φ be a homomorphism of K into L . If A is a (T, S)- intuitionistic fuzzy subalgebra on K then φ (A) is a (T, S)-intuitionistic fuzzy subalgebra on L . On the otherhand, if B is a (T, S)-intuitionistic fuzzy subalgebra of L then φ^-1 (B) is a (T, S)- intuitionistic fuzzy subalgebra of K .

Proof. Let f_α ∈ F and for every n (α)-tuples (x₁, x₂,. . . , x_n(α)) ∈ X^n(α), $\begin{matrix} φ^{- 1} (B) & (f_{α} (x_{1}, x_{2}, . . ., x_{n (a)})) \\ = & B (φ (f_{α} (x_{1}, x_{2}, . . ., x_{n (a)}))) \\ = & B (f_{α} (φ (x_{1}), φ (x_{2}), . . ., φ (x_{n (a)}))) \\ \geq & (T_{i = 1}^{n} μ_{B} (φ (x_{i})), S_{i = 1}^{n} ν_{B} (φ (x_{i}))) \\ = & (T_{i = 1}^{n} μ_{φ^{- 1} (B)} (x_{i}), S_{i = 1}^{n} ν_{φ^{- 1} (B)} (x_{i})) \end{matrix}$ then φ^-1 (B) is a (T, S)- intuitionistic fuzzy subalgebra of K . φ (A) can be proved simply.□

Proposition 2.4. Let K = [X, F] be an algebra. If A is a (T, S)-intuitionistic fuzzy subalgebra on K then so are □A and ◇A .

Proof. From definiton $ω_{f_{α}}^{(T, S)} (A_{1}, A_{2}, . . ., A_{n (α)}) ⊑ A$ and so, $\begin{matrix} sup_{f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}) = x} (T_{i = 1}^{n} μ_{A_{i}} (x_{i}), S_{i = 1}^{n} ν_{A_{i}} (x_{i})) \leq A (x) \end{matrix}$ for all (x₁,. . . , x_n(α)) ∈ X^n(α) . Since $inf_{f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}) = x} S_{i = 1}^{n} ν_{A_{i}} (x_{i}) \geq ν_{A} (x)$ then $inf_{f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}) = x} S_{i = 1}^{n} (1 - (1 - ν_{A_{i}} (x_{i}))) \geq 1 - (1 - ν_{A} (x))$ $\Rightarrow 1 - (inf_{f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}) = x} S_{i = 1}^{n} (1 - (1 - ν_{A_{i}} (x_{i})))) \leq 1 - ν_{A} (x)$ $\Rightarrow sup_{f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}) = x} (1 - (S_{i = 1}^{n} (1 - (1 - ν_{A_{i}} (x_{i}))))) \leq 1 - ν_{A} (x)$ $\Rightarrow sup_{f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}) = x} T_{i = 1}^{n} (1 - ν_{A_{i}} (x_{i})) \leq 1 - ν_{A} (x)$ . ◇ A is a (T, S)-intuitionistic fuzzy subalgebra. Now, □ A can be proved similarly.□

Theorem 2.5. Let K = [X, F] be an algebra. If A is a (T, S)-intuitionistic fuzzy subalgebra on K then A_T,1 is a crisp algebra of X .

Proof. Let f_α ∈ F and x₁, x₂,. . . , x_n(α) ∈ A_T,1 .For all f_α (x₁, x₂,. . . , x_n(α)) = x then $\begin{matrix} T (μ_{A} (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)})), \\ 1 - ν_{A} (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}))) \\ \geq & T (T_{i = 1}^{n} μ_{A} (x_{i}), 1 - (S_{i = 1}^{n} ν_{A} (x_{i}))) \\ = & T (T_{i = 1}^{n} μ_{A} (x_{i}), 1 - T_{i = 1}^{n} (1 - ν_{A} (x_{i}))) \\ = & T (T (μ_{A} (x_{n}), T_{i = 1}^{n} - 1 μ_{A} (x_{i})), \\ T (1 - ν_{A} (x_{n}), T_{i = 1}^{n} - 1 (1 - ν_{A} (x_{i})))) \\ ⋮ \\ = & T (T (μ_{A} (x_{n}), 1 - ν_{A} (x_{n})), . . ., \\ T (μ_{A} (x_{1}), 1 - ν_{A} (x_{1}))) \\ = & T (1, 1, . . ., 1) = 1 \end{matrix}$ So, f_α (x₁, x₂,. . . , x_n(α)) ∈ A_T,1 .

Theorem 2.6. Let K = [X, F] be an algebra. If A is a (T_M, S_M)- intuitionistic fuzzy subalgebra on K then for every r ∈ [0, 1], A_{T_M,r} is a crisp algebra of X .

Proof. Let f_α ∈ F and x₁, x₂,. . . , x_n(α) ∈ A_{T_M,r} . For all f_α (x₁, x₂,. . . , x_n(α)) = x then $\begin{matrix} T_{M} (μ_{A} (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)})), \\ 1 - ν_{A} (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}))) \\ \geq & T_{M} ({T_{M}}_{i = 1}^{n} μ_{A} (x_{i}), 1 - ({S_{M}}_{i = 1}^{n} ν_{A} (x_{i}))) \\ = & T_{M} ({T_{M}}_{i = 1}^{n} μ_{A} (x_{i}), 1 - {T_{M}}_{i = 1}^{n} (1 - ν_{A} (x_{i}))) \\ = & T_{M} (T_{M} (μ_{A} (x_{n}), {T_{M}}_{i = 1}^{n} - 1 μ_{A} (x_{i})), \\ T_{M} (1 - ν_{A} (x_{n}), {T_{M}}_{i = 1}^{n} - 1 (1 - ν_{A} (x_{i})))) \\ ⋮ \\ = & T_{M} (T_{M} (μ_{A} (x_{n}), 1 - ν_{A} (x_{n})), . . ., \\ T_{M} (μ_{A} (x_{1}), 1 - ν_{A} (x_{1}))) \\ \geq & T_{M} (r, r, . . ., r) = r \end{matrix}$ So, f_α (x₁, x₂,. . . , x_n(α)) ∈ A_{T_M,r} .□

Theorem 2.7. Let K = [X, F] be an algebra. If A is a (T, S)-intuitionistic fuzzy subalgebra on K then A_T,S,1 is a crisp algebra of X .

Proof. Let f_α ∈ F and x₁, x₂,. . . , x_n(α) ∈ A_T,S,1 . Then, T (μ_A (x_i) , S (μ_A (x_i) , ν_A (x_i))) = 1, i = 1, 2,. . . , n (α) . For all f_α (x₁, x₂,. . . , x_n(α)) = x, $\begin{matrix} T (μ_{A} (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)})), \\ S (μ_{A} (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}), ν_{A} (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)})))) \\ T (μ_{A} (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)})), μ_{A} (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}))) \\ \geq T (T_{i = 1}^{n} μ_{A} (x_{i}), T_{i = 1}^{n} μ_{A} (x_{i})) \\ = T (T (μ_{A} (x_{n}), T_{i = 1}^{n} - 1 μ_{A} (x_{i})), T (μ_{A} (x_{n}), T_{i = 1}^{n} - 1 μ_{A} (x_{i}))) \\ T (μ_{A} (x_{n}), μ_{A} (x_{n})), T (μ_{A} (x_{n - 1}), μ_{A} (x_{n - 1})), . . ., \\ T (μ_{A} (x_{1}), μ_{A} (x_{1}))) \\ = T (1, 1, . . ., 1) = 1 \end{matrix}$ So, f_α (x₁, x₂,. . . , x_n(α)) ∈ A_T,S,1 .□

Theorem 2.8. Let K = [X, F] be an algebra. If A is a (T_M, S_M)- intuitionistic fuzzy subalgebra on K then for every r ∈ [0, 1] , A_T,S,r is a crisp algebra of X .

Proof. It is clear.

2.1 (T, S)-Intuitionistic Fuzzy Congruence Relations

In this section, we defined congruence relations on (T, S)- intuitionistic fuzzy algebra and we examined some basic properties of (T, S)- intuitionistic fuzzy congruence relations.

Definition 2.3. Let K = [X, F] be an algebra and f_α ∈ F . For any (A₁, A₂,. . . , A_n(α)) ∈ IFR (X) ^n(α), $ϖ_{f_{α}}^{(T, S)} (A_{1}, A_{2}, . . ., A_{n (α)})$ to be an element of IFR (X) defined by $\begin{matrix} ϖ_{f_{α}}^{(T, S)} (A_{1}, A_{2}, . . ., A_{n (α)}) (x, y) \\ = sup (T_{i = 1}^{n} μ_{A_{i}} (x_{i}, y_{i}), S_{i = 1}^{n} ν_{A_{i}} (x_{i}, y_{i})) \end{matrix}$ where the supremum is taken over all representations of f_α (x₁, x₂,. . . , x_n(α)) = x and f_α (y₁, y₂,. . . , y_n(α)) = y . If no such n (α)-tuples for x and y then $ϖ_{f_{α}}^{(T, S)} (A_{1}, A_{2}, . . ., A_{n (α)}) (x, y) = Θ = (0, 1) .$

IFR (X) is (T, S)-intuitionistic fuzzy algebra on intuitionistic fuzzy relations with corresponding operations $ϖ_{f_{α}}^{(T, S)}$ on IFR (X) .

Definition 2.4. Let K = [X, F] be an algebra. A ∈ IFE (X) is said to be a (T, S)-intuitionistic fuzzy congruence relation on K if and only if, for each f_α ∈ F, $\begin{matrix} ϖ_{f_{α}}^{(T, S)} (A, A, . . ., A) ⊑ A . \end{matrix}$

Theorem 2.9. Let K = [X, F] be an algebra, f_α ∈ F and A₁, A₂,. . . , A_n(α), A be intuitionistic fuzzy relations on K . $\begin{matrix} ϖ_{f_{α}}^{(T, S)} (A_{1}, A_{2}, . . ., A_{n (α)}) ⊑ A \Leftrightarrow \\ A (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}), f_{α} (y_{1}, y_{2}, . . ., y_{n (α)})) \\ \geq (T_{i = 1}^{n} μ_{A} (x_{i}, y_{i}), S {i = 1}^{n} ν_{A} (x_{i}, y_{i})) \end{matrix}$ for all pairs of n (α)-tuples (x₁, x₂,. . . , x_n(α)) and (y₁, y₂,. . . , y_n(α)) .

Proof. The proof of this theorem can be seen easily from Theorem 2.1.□

Proposition 2.10. Let K = [X, F] be an algebra. If A is a (T, S)-intuitionistic fuzzy congruence relation on K then A_(r,s) (shortly ∼) is a crisp congruence relation for all r ∈ Δ_T and s ∈ Δ_S with r + s ≤ 1 .

Proof. Since A ∈ IFE (X) then A_(r,s) (shortly ∼) is a crisp equivalence relation on X. Now, we should prove substitution property. Let f_α ∈ F and f_α (x₁, x₂,. . . , x_n(α)) = x, f_α (y₁, y₂,. . . , y_n(α)) = y such that x, y ∈ X and (x₁, x₂,. . . , x_n(α)) , (y₁, y₂,. . . , y_n(α)) ∈ X^n(α) . If x_i ∼ y_i, i = 1, 2,. . . , n (α) then A (x_i, y_i) ≥ (r, s) . $\begin{matrix} A (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}), f_{α} (y_{1}, y_{2}, . . ., y_{n (α)})) \\ \geq sup_{x, y} (T_{i = 1}^{n} μ_{A} (x_{i}, y_{i}), S_{i = 1}^{n} ν_{A} (x_{i}, y_{i})) \\ \geq sup_{x, y} (T (μ_{A} (x_{n}, y_{n}), T_{i = 1}^{n} - 1 μ_{A} (x_{i}, y_{i})), \\ S (ν_{A} (x_{n}, y_{n}), S_{i = 1}^{n} - 1 ν_{A} (x_{i}, y_{i}))) \\ \geq sup_{x, y} (\begin{matrix} T (μ_{A} (x_{n}, y_{n}), T (μ_{A} (x_{n - 1}, y_{n - 1}), T_{i = 1}^{n} - 2 μ_{A} (x_{i}, y_{i}))), \\ S (ν_{A} (x_{n}, y_{n}), S (ν_{A} (x_{n - 1}, y_{n - 1}), S_{i = 1}^{n} - 2 ν_{A} (x_{i}, y_{i}))) \end{matrix}) \\ . . . \\ sup_{x, y} (\begin{matrix} T (μ_{A} (x_{n}, y_{n}), μ_{A} (x_{n - 1}, y_{n - 1}), . . ., μ_{A} (x_{1}, y_{1})), \\ S (ν_{A} (x_{n}, y_{n}), ν_{A} (x_{n - 1}, y_{n - 1}), . . ., ν_{A} (x_{1}, y_{1})) \end{matrix}) \\ \geq (T (r, r, . . ., r), S (s, s, . . ., s)) = (r, s) \\ \Rightarrow A (f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}), f_{α} (y_{1}, y_{2}, . . ., y_{n (α)})) \geq (r, s) \end{matrix}$ So, $\begin{matrix} f_{α} (x_{1}, x_{2}, . . ., x_{n (α)}) \sim f_{α} (y_{1}, y_{2}, . . ., y_{n (α)}) . \end{matrix}$ □

Proposition 2.11. Let K = [X, F], L = [Y, F] be two similar algebras and φ be a homomorphism of K into L . If A is a (T, S)- intuitionistic fuzzy congruence relation on K then φ (A) is a (T, S)-intuitionistic fuzzy congruence relation on L . If B is a (T, S)-intuitionistic fuzzy congruence relation on L then φ^-1 (B) is a (T, S)-intuitionistic fuzzy congruence relation on K .

Proof. It is clear.□

3 Conclusions

The (T, S)-intuitionistic fuzzy algebra was introduced. Thus, the concept of intuitionistic fuzzy algebra was extended. Sub algebraic structures and their properties were examined. Also, congruence relations over (T, S)-intuitionistic fuzzy algebras were studied. Some results were obtained with the level sets of these algebraic structures.

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