( T,S ) – Intuitionistic fuzzy algebras

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 A = { < x , μ A ( x ) , ν A ( x ) > : x ∈ X } 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: μ A ( x ) + ν A ( x ) ≤ 1 , forallx ∈ X .

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 A c = { < x , ν A ( x ) , μ A ( x ) > : x ∈ X }

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 A ( r , s ) = { x : x ∈ X s u c h t h a t μ A ( x ) ≥ r , v A ( x ) ≤ s } A 〈 r , s 〉 = { x : x ∈ X s u c h t h a t μ A ( x ) > r , v A ( x ) < s 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, μ R ( x , x ) = 1 ν R ( x , x ) = 0 then R is called an intuitionistic fuzzy reflexsive.

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 . [ a ] R = { < x , μ [ a ] R ( x ) , ν [ a ] R ( x ) > : x ∈ X } 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

f ( A ) ( y ) = ( μ f ( A ) ( y ) , ν f ( A ) ( y ) ) where μ f ( A ) ( y ) = { ⋁ { μ A ( x ) : x ∈ f - 1 ( y ) } 0 ; otherwise and ν f ( A ) ( y ) = { ⋀ { ν A ( x ) : x ∈ f - 1 ( y ) } 1 ; otherwise 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 f - 1 ( B ) ( x ) = ( μ f - 1 ( B ) ( x ) , ν f - 1 ( B ) ( x ) ) 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), f α ′ ( φ ( x 1 ) , φ ( x 2 ) , . . . , φ ( x n ( a ) ) ) = φ ( f α ( x 1 , x 2 , . . . , x n ( a ) ) ) .

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) , a i ≡ b i ( θ ) ⇒ f α ( a 1 , a 2 , . . . , a n ( a ) ) ≡ f α ( b 1 , b 2 , . . . , b n ( a ) ) ( θ ) .

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; ω α : IFS ( X ) × IFS ( X ) × . . . × IFS ( X ) → IFS ( X ) , ω α ( A 1 , A 2 , . . . , A n ( α ) ) = A such that A ( x ) = { sup { A 1 ( x 1 ) ∧ . . . ∧ A n ( α ) ( x n ( α ) ) } , f α ( x 1 , . . . , x n ( α ) ) = x Θ = ( 0 , 1 ) , otherwise 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: ω α ( A ) = A e A e ( x ) = { sup x ∈ X A ( x ) , & x = e ( 0 , 1 ) , x ≠ e

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

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 ) = { 0 , if ( x , y ) ∈ [ 0 , 1 ) 2 min ( x , y ) , otherwise (drastic product)

Definition 1.15. [19] For n ∈ ℕ , the function T n : ∏ i ∈ I n [ 0 , 1 ] → [ 0 , 1 ] define by 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 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

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 ) = { 1 , if ( x , y ) ∈ ( 0 , 1 ] 2 max ( x , y ) , otherwise (drastic product)

It is clear that, T ( x , y ) ≤ min ( x , y ) ≤ max ( x , y ) ≤ S ( x , y ) 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; Δ T = { x ∈ [ 0 , 1 ] : T ( x , x ) = x } Δ S = { x ∈ [ 0 , 1 ] : S ( x , x ) = x }

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 S ( x , y ) = 1 - T ( 1 - x , 1 - y ) .

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 ∈ ℕ , the function S n : ∏ i ∈ I n [ 0 , 1 ] → [ 0 , 1 ] define by 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 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: S i = 1 n x i = 1 - T i = 1 n ( 1 - x i )

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 A T , r = { x ∈ X : T ( μ A ( x ) , 1 - ν A ( x ) ) ≥ r } 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 A T , S , r = { x ∈ X : T ( μ A ( x ) , S ( μ A ( x ) , ν A ( x ) ) ) ≥ r } 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 _M and 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; ω f α ( T , S ) : IFS ( X ) × . . . × IFS ( X ) → IFS ( X ) , ω f α ( T , S ) ( A 1 , A 2 , . . . , A n ( α ) ) = A , A ( x ) = { 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 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: ω f α ( T , S ) ( A ) = A e A e ( x ) = { sup x ∈ X A ( x ) , x = e ( 0 , 1 ) , x ≠ e

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. ω f α ( T , S ) ( A 1 , A 2 , . . . , A n ( α ) ) ⊑ A if and only if A ( f α ( x 1 , x 2 , . . . , x n ( α ) ) ) ≥ ( T i = 1 n μ A i ( x i ) , S i = 1 n ν A i ( x i ) ) is true for every (x₁, x₂,. . . , x_n(α)) ∈ X^n(α) .

Proof. (1) Let n (α) ≠0 and 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 ) ≤ A ( x ) , for all x ∈ X . So, 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 ) ) ≤ A ( x ) , for all (x₁,. . . , x_n(α)) ∈ X^n(α) . A ( f α ( x 1 , x 2 , . . . , x n ( α ) ) ) ≥ ω f α ( T , S ) ( A 1 , A 2 , . . . , A n ( α ) ) ( f α ( x 1 , x 2 , . . . , x n ( α ) ) ) = 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 ) ) ≥ ( T i = 1 n μ A i ( x i ) , S i = 1 n ν A i ( x i ) ) Conversely, let A ( f α ( x 1 , x 2 , . . . , x n ( α ) ) ) ≥ ( 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 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 ) ) ≤ 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 ) ) ≤ A ( x ) . 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 ) ≤ 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, ω f α ( T , S ) ( B ) ( x ) ≤ A ( x ) ↔ A ( e ) ≥ ω f α ( T , S ) ( B ) = x _ sup B ( x ) ⇔ A ( ω f α ( T , S ) ( x ) ) ≥ B ( x ) for all x ∈ X .□

Example 3. Let consider ℤ 6 algebra which defined with two binary operations (+ , ·), two nullary operations ( 0 ¯ , 1 ¯ ) and a unary operation (-). The intuitionistic fuzzy set A ∈ IFS ( ℤ 6 ) defined by A = { ( 0 ¯ , 0.5 , 0.1 ) , ( 1 ¯ , 0.2 , 0.5 ) , ( 2 ¯ , 0.3 , 0.3 ) , ( 3 ¯ , 0.4 , 0.4 ) , ( 4 ¯ , 0.3 , 0.3 ) , ( 5 ¯ , 0.2 , 0.5 ) } 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 A = ∩ i ∈ Λ A i 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) . A ( f α ( x 1 , x 2 , . . . , x n ( α ) ) ) = ⋂ i ∈ Λ A i ( f α ( x 1 , x 2 , . . . , x n ( α ) ) ) ≥ ( inf i ∈ I T j = 1 n μ A i ( x j ) , sup i ∈ I S j = 1 n ν A i ( x j ) ) ≥ ( T j = 1 n ( inf i ∈ I μ A i ( x j ) ) , S j = 1 n ( sup i ∈ I ν A i ( x j ) ) ) = ( T j = 1 n μ A ( x j ) , S j = 1 n ν A ( x j ) ) 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), 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 ) ) ) ) ≥ ( 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 ) ) 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, 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 ) ) ≤ A ( x ) 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 ) ≥ 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 ) ) ) ≥ 1 - ( 1 - ν A ( x ) ) ⇒ 1 - ( inf f α ( x 1 , x 2 , . . . , x n ( α ) ) = x S i = 1 n ( 1 - ( 1 - ν A i ( x i ) ) ) ) ≤ 1 - ν A ( x ) ⇒ sup f α ( x 1 , x 2 , . . . , x n ( α ) ) = x ( 1 - ( S i = 1 n ( 1 - ( 1 - ν A i ( x i ) ) ) ) ) ≤ 1 - ν A ( x ) ⇒ sup f α ( x 1 , x 2 , . . . , x n ( α ) ) = x T i = 1 n ( 1 - ν A i ( x i ) ) ≤ 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 T ( μ A ( f α ( x 1 , x 2 , . . . , x n ( α ) ) ) , 1 - ν A ( f α ( x 1 , x 2 , . . . , x n ( α ) ) ) ) ≥ 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 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 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 ) ) ) ) ≥ 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 ) ) ) ≥ T M ( r , r , . . . , r ) = r 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, 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 ) ) ) ) ≥ 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 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.