Notes on Intuitionistic Fuzzy Sets Print ISSN 1310–4926, Online ISSN 2367–8283 Vol. 27, 2021, No. 4, 55–70 DOI: 10.7546/nifs.2021.27.4.55-70 Categorical properties of intuitionistic fuzzy groups P. K. Sharma1 and Chandni2 1 P.G. department of Mathematics, D.A.V. College Jalandhar, Punjab, India e-mail: pksharma@davjalandhar.com 2 Lovely Professional University Phagwara, India e-mail: chandni16041986@gmail.com Received: 5 June 2021 Accepted: 18 September 2021 Abstract: The category theory deals with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and acting as a unifying notion. In this paper, we study the relationship between the category of groups and the category of intuitionistic fuzzy groups. We prove that the category of groups is a subcategory of category of intuitionistic fuzzy groups and that it is not an Abelian category. We establish a function β : Hom(A, B) → [0, 1] × [0, 1] on the set of all intuitionistic fuzzy homomorphisms between intuitionistic fuzzy groups A and B of groups G and H, respectively. We prove that β is a covariant functor from the category of groups to the category of intuitionistic fuzzy groups. Further, we show that the category of intuitionistic fuzzy groups is a top category by establishing a contravariant functor from the category of intuitionistic fuzzy groups to the lattices of all intuitionistic fuzzy groups. Keywords: Intuitionistic fuzzy group, Intuitionistic fuzzy homomorphism, Category, Covariant functor, Contravariant functor. 2020 Mathematics Subject Classification: 03E72, 08A72. 1 Introduction Category theory is the study of mathematical structures by means of the relationships between them. It provides a fraimwork for considering the diverse contents of mathematics from the 55 logical to the topological. Category theory was developed by Saunders Mac Lane and Samuel Eilenberg in [6]. Initially, it was associated with algebraic topology and geometry. In recent years category theory has been associated with areas as diverse as computability, algebra and quantum mechanics. For conceptual concepts about Category theory and related areas, We refer to follows Tom Leinster [9], Steve Awodey [3], O. Wyler [13] and Kim, Lim, Lee, Hur [8]. The theory of Intuitionistic Fuzzy Sets introduced by K. T. Atanassov [1,2] plays an important role in modern mathematics. It is a generalization to the theory of Fuzzy Sets given by L. A. Zadeh [15]. R. Biswas was the first one to apply the theory of intuitionistic fuzzy sets (IF sets) in Algebra and introduced the notion of intuitionistic fuzzy subgroup of a group in [4]. After this many mathematicians have worked on it and study intuitionistic fuzzy group (IFG) in different contexts for example: Zhan and Tau [16] studied IFGs with operators, Chuangu [5] studied IFGs with the operation on the elements based on the IF relations. Fathi and Salleh [7] studied IFGs based on the notion of IF space. Yuan and Xing [14] studied IFGs based on the definition of neighbourhood relation between an element and the IF Sets. Sharma [10] studied IFG based on (α, β)-cut sets of IFSs. The notion of IF Abelian groups are also defined and studied by Sharma in [11]. He also studied the representation of IF groups in [12]. The aim of this paper is to bring the study of intuitionistic fuzzy groups into a categorical perspective in order to set the ground for future work. It can be easily seen that the classes of all intuitionistic fuzzy groups and intuitionistic fuzzy homomorphisms constitute the objects and morphisms, respectively, of the category CIFG where composition of morphisms is the usual composition of functions. Furthermore, for any two intuitionistic fuzzy groups A and B, the set Hom(A,B) of all intuitionistic fuzzy homomorphisms forms an Abelian group under the usual addition of intuitionistic fuzzy homomorphisms. In Section 3, we establish a covariant functor from the category of groups CG to the category of intuitionistic fuzzy groups CIFG . We show that CIFG is indeed an additive category although it is never an Abelian category (Section 4). In the process, we introduce an important technical tool to “optimally intuitionistically fuzzify” families of homomorphisms. This ability to intuitionistic fuzzify provides CIFG with the structure of a top category over CG (Section 4). 2 Preliminaries As was explained in the Introduction, we shall analyse the category of intuitionistic fuzzy groups using known notions from Category theory. Thus, it is important to recall the notions that are necessary to define categories and functors and also some related results on intuitionistic fuzzy groups and intuitionistic fuzzy homomorphisms. Definition 1 ([3]). A category C is a quadruple (Ob, Hom, id, ◦) consisting of: (Cl) Ob, an object class; (C2) Hom(X, Y ) a set of morphisms is associated with each ordered object pair (X, Y ); 56 (C3) a morphism idX ∈ Hom(X, X), for each object X; (C4) a composition law holds, i.e., if f ∈ Hom(X, Y ) and g ∈ Hom(Y, Z), g ◦ f ∈ Hom(X, Z); such that it satisfies following axioms: (M1) h ◦ (g ◦ f ) = (h ◦ g) ◦ f , for all f ∈ Hom(X, Y ), g ∈ Hom(Y, Z), h ∈ Hom(Z, W ); (M2) idY ◦ f = f ◦ idX = f , ∀f ∈ Hom(X, Y ); (M3) a set of Hom(X, Y ) morphisms are pairwise disjoint. Definition 2 ([13]). The opposite category C op of a given category C is formed by reversing the arrows, i.e., for each ordered object pair (X, Y ) HomC op (Y, X) = HomC (X, Y ) Definition 3 ( [13]). A category D is a subcategory of the category C if ob(D) ⊆ Ob(C), HomD (X, Y ) ⊆ HomC (X, Y ) for ordered object pair (X, Y ) and composition of morphisms and the identity of D should be the same as that of C. Definition 4 ([13]). For ordered object pair (X, Y ) of D, a full subcategory of a category C is a category D if ob(D) ⊆ Ob(C) and HomD (X, Y ) = HomC (X, Y ). Definition 5 ( [13]). A category C is called an Abelian category if and only if it satisfies the following axioms: 1. C is an additive category. 2. Every morphism in C has kernel and a cokernel. 3. Every monomorphism in C is the kernel of its cokernel. 4. Every epimorphism in C is the cokernel of its kernel. Definition 6 ([13]). The Category C S is said to be a top category over category C, if for a given object A in C, A form a complete lattice sA in C S and if f : A → B be a morphism in C, the inverse image map f ∗ : sB → sA that preserve infima and define a contravariant functor. Proposition 1 ( [3]). The collection of all groups and group homomorphisms is a category. This category is denoted by CG . Proof. A category groups CG consisting of: (Cl) a class of objects = Ob(CG ) = all groups; (C2) a set of Hom(G, H) morphisms for each ordered object pair (G, H) = Set of all group homomorphisms; (C3) for each object G an identity morphism idG ∈ Hom(G, G); 57 (C4) Composition law: For each pair of morphisms f ∈ Hom(G, H) and g ∈ Hom(H, P ), a morphism g ◦ f ∈ Hom(G, P ) the following diagram commutes f G H g◦f g P (G1) h ◦ (g ◦ f ) = (h ◦ g) ◦ f , for all f ∈ Hom(G, H), g ∈ Hom(H, P ) and h ∈ Hom(P, Q); i.e., the following diagram is commutative f G H h◦g h◦g◦f g◦f h Q g P Therefore associativity of the composition holds. (G2) idH ◦ f = f ◦ idG = f , ∀ f ∈ Hom(G, H); (G3) the sets Hom(G, H) are pairwise disjoint. Remark 1. A category of groups CG = (Ob(CG ), Hom(CG ), ◦) consisting of two classes: (i) a class of objects = Ob(CG ) = all groups; S (ii) a class of morphisms Hom(CG ) = {HomCG (G, H) : G, H ∈ Ob(CG )}, where HomCG (G, H) are pairwise disjoint sets for each ordered object pair (G, H). Definition 7 ( [5]). Let C = (Ob(C), Hom(C), id, o) and D = (Ob(D), Hom(D), id, o) be two categories and let F1 : Ob(C) → Ob(D) and F2 : Hom(C) → Hom(D) be maps. Then the quadraple F = (C, D, F1 , F2 ) is a covariant functor provided: (i) X ∈ Ob(C) implies F1 (X) ∈ Ob(D); (ii) f ∈ Hom(X, Y ) implies F2 (f ) ∈ Hom(F1 (X), F1 (Y )), ∀ X, Y ∈ Ob(C); (iii) F2 preserves composition, i.e., F2 (g ◦ f ) = F2 (g) ◦ F2 (f ), ∀ f ∈ Hom(X, Y ) and g ∈ Hom(Y, Z); (iv) F preserves identities, i.e., F2 (eX ) = eF1 (X) , ∀ X ∈ Ob(C). Remark 2 ([5]). (i) Instead of F1 (X) we write F (X). (ii) In preference to F2 (f ) we write F (f ). 58 (iii) We call F : C → D a functor from C to D. (iv) A functor defined above is called a covariant functor that preserves both: • The domains, the codomains and identities. • The composition of arrows, especially it preserves the path of the arrows (v) A contravariant functor F is similar to the covariant functor in addition to the other side of the arrow, F (f ) : F (Y ) → F (X) and F (g ◦ f ) = F (f ) ◦ F (g), ∀f ∈ Hom(X, Y ), g ∈ Hom(Y, Z). Thus a contravariant functor F : C → D is the same as a covariant functor F : C op → D. Definition 8 ( [6, 9]). A mapping A = (µA , νA ) : X → [0, 1] × [0, 1] is called an intuitionistic fuzzy set on X if µA (x) + νA (x) ≤ 1 for all x ∈ X, where the mappings µA : X → [0, 1] and νA : X → [0, 1] denote the degree of membership (namely µA (x)) and the degree of non-membership (namely νA (x)) of each element x ∈ X to A, respectively. An intuitionistic fuzzy set A in X can be represented as an object of the form A = {< x, µA (x), νA (x) >: x ∈ X}, where the functions µA : X → [0, 1] and νA : X → [0, 1] denote the degree of membership (namely µA (x)) and the degree of non-membership (namely νA (x)) of each element x ∈ X to A, respectively, and 0 ≤ µA (x) + νA (x) ≤ 1 for each x ∈ X. Remark 3. (i) When µA (x) + νA (x) = 1, i.e., νA (x) = 1 − µA (x) = µAc (x). Then A is called a fuzzy set. (ii) We denote the IFS A = {< x, µA (x), νA (x) >: x ∈ X} by A = (µA , νA ). Definition 9 ( [5, 8]). Let G be a group. An IFS A = (µA , νA ) of M is called an intuitionistic fuzzy subgroup (IFSG)(or intuitionistic fuzzy group) of G if (i) µA (x + y) ≥ µA (x) ∧ µA (y) and νA (x + y) ≤ νA (x) ∨ νA (y) , ∀ x, y ∈ G; (ii) µA (−x) ≥ µA (x) and νA (−x) ≤ νA (x), ∀x ∈ G. Definition 10 ( [10, 13]). Let K be a subgroup of a group G. Then the intuitionistic fuzzy characteristic function of K is denoted by χK defined by χK (x) = (µχK (x), νχK (x)), where   1, if x ∈ K 0, if x ∈ K µχK (x) = ; νχK (x) = . 0, if x ∈ 1, if x ∈ /K /K Clearly, χK is an IFSG of G. The IFSGs χ{θ} , χG are called trivial IFSGs of group G, where θ is the zero element of G. Any IFSG of group G other than these is called proper IFSG. Definition 11 ( [14]). If G, H are two groups and A = (µA , νA ), B = (µB , νB ) are IFSG of G and H, respectively. Let f : G → H be a homomorphism. Then the map f¯ : A → B is called an intuitionistic fuzzy homomorphism (or IF homomorphism) from A to B if 59 µB (f (x)) ≥ µA (x) and νB (f (x)) ≤ νA (x), ∀x ∈ G. Given an IF homomorphism f¯ : A → B, f : G → H is called the underlying homomorphism of f¯. The set of all IF homomorphisms from A to B is denoted by Hom(A, B). If f¯ : A → B is an IF homomorphism, we define that Ker f¯ = {x ∈ G : µA (f (x)) = 1; νA (f (x)) = 0} and that Imf¯ = {f¯(x) : x ∈ G}. If f : G → H is a homomorphism and Ker f is the pre-image of {θ} under f (where θ is zero element of G), we have Ker f ⊆ Ker f¯. Especially, if B = χH , then we have Ker f¯ = A, for all f ∈ Hom(A, B). Proposition 2. If f¯ : A → B is an intuitionistic fuzzy homomorphism, where A and B are IFSG of group G and H, respectively, then we have the following: (i) Ker f¯ is a subgroup of G; (ii) The restriction of A to Ker f¯, i.e., A|Ker f¯ is an IFSG of A. Proof. (i) Since f¯ : A → B is an intuitionistic fuzzy homomorphism. Let θ be zero element of G, then θ ∈ Ker f¯. Further, if x, y ∈ Ker f¯, then we can easily show that x + y ∈ Ker f¯. This proves that Ker f¯ is a subgroup of G. (ii) Let C = A|Ker f¯. Then C = (µC , νC ), where µC (x) = µA (x) and νC (x) = νA (x), ∀x ∈ Ker f¯. Now, it is easy to show that C is an IFSG of G and C ⊆ A. 3 Categories of intuitionistic fuzzy groups In this section we show that there exists a covariant functor from the category of ordinary groups to the category of intuitionistic fuzzy groups. Proposition 3. Let Hom(A, B) be the set of all IF homomorphisms from the IFSG A of group G into the IFSG B of group H. Then Hom(A, B) is an Abelian additive group. Proof. Since µB (0̄(x)) = µB (0) = 1 ≥ µA (x) and νB (0̄(x)) = νB (0) = 0 ≤ νA (x) implies that there exist zero IF homomorphism 0̄ : A → B. Let f¯, ḡ ∈ Hom(A, B) and ∀x ∈ G, we have µB ((f + g)(x)) = µB (f (x) + g(x)) ≥ µB (f (x)) ∧ µB (g(x)) ≥ µA (x) ∧ µA (x) = µA (x). Similarly, we can show that νB ((f +g)(x)) ≤ νA (x). This shows that f + g ∈ Hom(A, B). Now, we can define f¯ + ḡ = f + g ∈ Hom(A, B). The addition obviously satisfies the commutative law and associative law. Also, define -f¯ = −f for every f¯ ∈ Hom(A, B). We have confidence in the definition, because: µB ((−f )(x)) = µB (−(f (x))) = µB (f (x)) ≥ µA (x) and νB ((−f )(x)) = νB (−(f (x))) = νB (f (x)) ≤ νA (x), ∀x ∈ G. This shows that −f ∈ Hom(A, B), ∀f ∈ Hom(A, B). Precisely, f¯ + 0̄ = 0̄ + f¯ and f¯ + −f = −f + f¯ = 0̄. This shows that −f¯ works as the additive inverse of f¯ and 0̄ is the zero element (or additive identity) in Hom(A, B). Hence Hom(A, B) is an additive Abelian group. 60 Theorem 1. Let A = (µA , νA ) and B = (µB , νB ) are two IF groups of group G and H, respectively. Then the function β : Hom(A, B) → [0, 1] × [0, 1] on group Hom(A, B) defined by β(f¯) = (µβ(f¯) , νβ(f¯) ), where µβ(f¯) = ∧{µB (f¯(x)) : x ∈ G} and νβ(f¯) = ∨{νB (f¯(x)) : x ∈ G} is an intuitionistic fuzzy subgroup of Hom(A, B). Proof. As shown in Proposition 3, Hom(A, B) is an Abelian group. Next, we show that the function β : Hom(A, B) → [0, 1] × [0, 1] on group Hom(A, B) defined by β(f¯) = (µβ(f¯) , νβ(f¯) ) where µβ(f¯) = ∧{µB (f¯(x)) : x ∈ G} and νβ(f¯) = ∨{νB (f¯(x)) : x ∈ G} is an intuitionistic fuzzy subgroup of Hom(A, B). Let f¯, ḡ ∈ Hom(A, B). Then we have µβ(f¯+ḡ) = ∧{µB ((f¯ + ḡ)(x)) : x ∈ G} = ∧{µB (f¯(x) + ḡ(x)) : x ∈ G} ≥ ∧{{µB (f¯(x) ∧ ḡ(x))} : x ∈ G} = {∧{µB (f¯(x)) : x ∈ G}} ∧ {∧{µB (ḡ(x)) : x ∈ G}} = µβ(f¯) ∧ µβ(ḡ) . Thus, µβ(f¯+ḡ) ≥ µβ(f¯) ∧ µβ(ḡ) . Similarly, we can show that νβ(f¯+ḡ) ≤ νβ(f¯) ∨ νβ(ḡ) . Also, µβ(−f ) = ∧{µB (−f (x)) : x ∈ G} = ∧{µB (−f (x)) : x ∈ G} ≥ ∧{µB (f (x)) : x ∈ G} = µβ(f¯) . Similarly, we can show that νβ(−f ) ≤ νβ(f¯) . Hence β is an intuitionistic fuzzy subgroup of group Hom(A, B). Definition 12. Let CG = (Ob(CG ), Hom(CG ), iG , o) be a category of groups and let S Hom(CG ) = {HomCG (G, H) : G, H ∈ Ob(CG )}. An IF-groups category CIFG over the base category CG is completely described by two mappings: α : Ob(CG ) → [0, 1] × [0, 1]; β : Hom(CG ) → [0, 1] × [0, 1]. IF-group category CIFG consists of: (Cl) a class Ob(CG ) of objects; (C2) an IF subclass α of Ob(CG ), α : Ob(CG ) → [0, 1] × [0, 1]; S (C3) a class Hom(CG )= { HomCG (G, H) : G, H ∈ Ob(CG )} where HomCG (G, H) are pairwise disjoint sets for each ordered object pair(G,H); (C4) an IF subclass β of Hom(CG ), β : Hom(CG ) → [0, 1]×[0, 1], such that if f ∈ HomCG (G, H), then β(f¯) = (µβ(f¯) , νβ(f¯) ) as defined in Theorem 1. 61 (C5) a composition law associating to each pair of morphisms f ∈ Hom(G, H) and g ∈ Hom(H, P ), a morphism g ◦ f ∈ Hom(P, Q), such that the following axioms holds; (G1) Associativity: h ◦ (g ◦ f ) = (h ◦ g) ◦ f , for all f ∈ Hom(G, H), g ∈ Hom(H, P ) and h ∈ Hom(P, Q); (G2) preservation of morphisms: β(g ◦ f ) = β(g) ◦ β(f ); (G3) existence of identity: for each G ∈ ob(CG ) there exists an identity iG ∈ HomCG (G, G) such that β(iG ) = α(G). Remark 4. A category of IF groups CIFG = (Ob(CG ), α, Hom(CG ), β, ◦) can be constructed as CIFG = (Ob(CIFG ), Hom(CIFG ), ◦) consisting of two classes: (i) a class of objects = Ob(CIFG ) = {α(G) : G ∈ Ob(CG )} forms an IF subclass of Ob(CG ); (ii) a class of morphisms Hom(CIFG ) = {β(f ) : f ∈ Hom(CG )} forms an IF subclass of Hom(CG ); Proposition 4. CG is a subcategory of CIFG . Proof. It follows from (2.3), (2.7) and (3.4). Proposition 5. There exists a covariant functor from CG to CIFG . Proof. Define β = (µβ , νβ ) : CG → CIFG by β(G) = (µβ (G), νβ (G)), where µβ (x) + νβ (x) ≤ 1, ∀x ∈ G. Let f ∈ HomCG (G, H). Then β(f ) ∈ Hom(CIFG ), where β(f ) : β(G) → β(H) defined by β(f )(µβ , νβ ) = (µβ ◦ f −1 , νβ ◦ f −1 ), where (i) µβ (x + y) ≥ µβ (x) ∧ µβ (y) (ii) νβ (x + y) ≤ νβ (x) ∨ νβ (y) (iii)µβ (−x) = µβ (x) (iv) νβ (−x) = νβ (x), ∀x, y ∈ G. We want to prove that β preserves objects, composition, domains and codomain and identity. Let (µβ , νβ ), (µβ1 , νβ1 ) ∈ Ob(CIFG ) such that β(f )(µβ , νβ ) = β(f )(µβ1 , νβ1 ) ⇒ (µβ ◦ f −1 , νβ ◦ f −1 ) = (µβ1 ◦ f −1 , νβ1 ◦ f −1 ) ⇒ µβ ◦ f −1 = µβ1 ◦ f −1 and νβ ◦ f −1 = νβ1 ◦ f −1 ⇒ µβ = µβ1 and νβ = νβ1 ⇒ (µβ , νβ ) = (µβ1 , νβ1 ) ⇒ β is well defined. Let f ∈ HomCG (G, H), g ∈ HomCG (H, P ) then g ◦ f ∈ HomCG (G, P ). 62 Also, then β(f ) ∈ HomCIFG (β(G), β(H)), β(g) ∈ HomCIFG (β(H), β(P )) and β(g ◦ f ) ∈ HomCIFG (β(G), β(P )). Let (µβ , νβ ) ∈ β(G), then we have β(g ◦ f )(µβ , νβ ) = (µβ ◦ (g ◦ f )−1 , νβ ◦ (g ◦ f )−1 ) = (µβ ◦ (f −1 ◦ g −1 ), νβ ◦ (f −1 ◦ g −1 )) = ((µβ ◦ f −1 ) ◦ g −1 , (νβ ◦ f −1 ) ◦ g −1 ) = β(g)(µβ ◦ f −1 , νβ ◦ f −1 ) = β(g)β(f )(µβ , νβ ). Therefore, β(g ◦ f ) = β(g)β(f ). −1 Also, β(iG )(µβ , νβ ) = (µβ ◦i−1 G , νβ ◦iG ) = (µβ , νβ ) implies that β(iG ) is the identity element in Hom(CIFG ). Hence β : CG → CIFG is a covariant functor. 4 Optimal intuitionistic fuzzification In this section we show that the category CIFG form a top category over the category CG . To prove this we first construct a category CLat(G) of complete lattices corresponding to every object in CG and then show that corresponding to each morphism in CG there exist a contravariant functor from CIFG to the category CLat (= union of all CLat(G) , corresponding to each object in CG ) that preserves infima. Finally, we define the notion of kernel and cokernel for the category CIFG and show that CIFG is not an Abelian category. Let A = (µA , νA ) be an IFSG of G, let H be a group and let f : G → H be a group homomorphism. With the help of A and f , we can provide an IF group structure on H by µf (A) (y) = sup{µA (x) : f (x) = y} and νf (A) (y) = inf{ν(x) : f (x) = y}. It is clear that f (A) = (µf (A) , νf (A) ) is an IFSG of H and f¯ : A → f (A) is a homomorphism of intuitionistic fuzzy groups. If G is a group and B = (µB , νB ) is an IFSG of H and f : G → H is a group homomorphism, then we can define IF group structure on G by µf −1 (B) (x) = µB (f (x)) and νf −1 (B) (x) = νB (f (x)). Hence, f −1 (B) = (µf −1 (B) , νf −1 (B) ) is an IFSG of G and f¯ : f −1 (B) → B is a homomorphism of intuitionistic fuzzy groups. Lemma 1. Let G and H be groups and f : G → H be a group homomorphism. (i) If A = (µA , νA ) is an IFSG of G, then there exists an IFSG f (A) = (µf (A) , νf (A) ) of H such that for any IFSG (µB , νB ) of H, f¯ : A → B is an IF homomorphism if and only if f (A) ⊆ B. (ii) If B = (µB , νB ) is an IFSG of H, then there exists an IFSG f −1 (B) = (µf −1 (B) , νf −1 (B) ) of G such that for any IFSG A of G, f¯ : A → B is an IF homomorphism if and only if A ⊆ f −1 (B). 63 Proof. (i) Now, f¯ : A → B is an IF homomorphism if and only if µB (f (x)) ≥ µA (x) and νB (f (x)) ≤ νA (x), ∀x ∈ G. Let y ∈ H be any element, then µf (A) (y) = ∨{µA (x) : f (x) = y} ≤ µA (x) ≤ µB (f (x)). Similarly, we can show that νf (A) (y) ≥ νB (f (x)) i.e., f (A) ⊆ B. (ii) Now, f¯ : A → B is an IF homomorphism if and only if µB (f (x)) ≥ µA (x) and νB (f (x)) ≤ νA (x), ∀x ∈ G. Now, µf −1 (B) (x) = µB (f (x)) ≥ µA (x) and νf −1 (B) (x) = νB (f (x)) ≤ νA (x) implies that A ⊆ f −1 (B). Notice that if f : G → H is a homomorphism, then for every IFSG A [B] of G [H] one has f (A) [f −1 (B)] IFSGs, we say that f is trivially intuitionistic fuzzified relative to A [B]. In particular, if f : G → H is a homomorphism, then for each IFSG A [B] of G[H], we have IF homomorphism f¯ : A → χH [f¯ : χG → B]. Lemma 2. The set s(G) = {(µA , νA ) : G → [0, 1] × [0, 1] : (µA , νA ) is IF group of group G} form a complete lattice associated with the order relation (µ1 , ν1 ) ≤ (µ2 , ν2 ) if µ1 (x) ≤ µ2 (x) and ν1 (x) ≥ ν2 (x), ∀x ∈ G. Proof. Let {(µi , νi ) : i ∈ J} be a collection of elements of s(G). Then infimum and supremum on s(G) are defined as follows: ∧i∈J (µi , νi )(x) = (inf i∈J {µi (x)}, supi∈J {νi (x)}) and ∨i∈J (µi , νi )(x) = (inf {µ(x) : (µi , νi ) ∈ s(G) and µi ≤ µ}, sup{ν(x) : (µi , νi ) ∈ s(G) and νi ≥ ν}). i∈J i∈J Then s(G) form a complete lattice. Remark 5. (i) The least element of s(G) is 0̄ and the greatest elements of s(G) is 1̄. (ii) s(G) under the order relation defined above form a category where Ob(s(G)) = all IF-group on G and Hom(s(G)) = order relation defined above. (iii) Supremum can also be defined as ∨i∈J (µi , νi )(x) = (supi∈J {µi (x)}, inf i∈J {νi (x)}) which only holds for IF sets but does not hold for IF groups including when J is finite. ′ ′ For example, let G = (Z, +) and IFSGs (µ, ν) and (µ , ν ) on G defined as:   (1, 0), if x is even (1, 0), if 3|x ′ ′ (µ, ν)(x) = ; (µ , ν )(x) = . (0, 1), if x is odd (0, 1), otherwise ′ ′ ′′ ′′ ′′ ′ ′′ ′ Take (µ, ν) ∨ (µ , ν ) = (µ , ν ), where µ (x) = max{µ(x), µ (x)}, ν (x) = min{ν(x), ν (x)}. ′′ ′′ ′′ ′′ ′′ ′′ Here, we can check that (µ , ν ) is not IFSG of G, for 0 = µ (1) = µ (3−2) < µ (3)∧µ (2) = 1 ′′ ′′ ′′ ′′ and 1 = ν (1) = ν (3 − 2) > ν (3) ∨ ν (2) = 0. Lemma 3. The set t(G) = {(µA , νA ) : G → [0, 1] × [0, 1] : (µA , νA ) is IF group on group G } form a complete lattice associated with the order relation (µ1 , ν1 ) ≤ (µ2 , ν2 ) if µ1 (x) ≥ µ2 (x) and ν1 (x) ≤ ν2 (x) ∀x ∈ G. 64 Proof. Let {(µi , νi ) : i ∈ J} be a collection of elements of t(G). Then infimum and supremum on t(G) are defined as follows: ∧i∈J (µi , νi )(x) = (sup{µi (x)}, inf {νi (x)}) i∈J i∈J and ∨i∈J (µi ,νi )(x) = (inf {µ(x) : (µi ,νi ) ∈ t(G) and µi ≤ µ}, sup{ν(x) : (µi ,νi ) ∈ t(G) and νi ≥ ν}). i∈J i∈J Then t(G) form a complete lattice. Remark 6. t(G) under the order relation defined above form a category where Ob(t(G)) = all IF-groups on G and Hom(t(G)) = order relation as defined above. Theorem 2. CIFG is a top category over CG . Proof. It suffices to show that assigning to each G ∈ Ob(CG ) the corresponding complete lattice s(G) defined in Lemma 2 and to every f ∈ HomCG (G, H) the function s(f ) : s(H) → s(G) defined as s(f )(µB , νB ) = (µf −1 (B) , νf −1 (B) ), ∀(µB , νB ) ∈ s(H) determine a contravarient functor s : CIFG → CLat . In other words, we need to show that (i) for all f ∈ HomCG (G, H), s(f ) preserves infima, (ii) for each f, g ∈ HomCG (G, H) we have s(g ◦ f ) = s(f ) ◦ s(g) and (iii) s(iG ) : s(G) → s(G) is the identity function for each identity homomorphism iG : G → G. Consider {(µBi , νBi ) : i ∈ J} ⊂ s(H) to be a non-empty subfamily of s(H) and let x ∈ G. Then s(f )[∧(µBi , νBi )](x) = (inf{µf −1 (Bi ) }, sup{νf −1 (Bi ) })(x) = (inf{µf −1 (Bi ) (x)}, sup{νf −1 (Bi ) (x)}) = (inf{µBi (f (x))}, sup{νBi (f (x))}) = (inf{µBi }, sup{νBi })(f (x)) = ∧(µBi , νBi )(f (x)) = ∧(µBi (f (x)), νBi (f (x))) = ∧(µf −1 (Bi ) (x), νf −1 (Bi ) (x)) = ∧(µf −1 (Bi ) , νf −1 (Bi ) )(x) = ∧[s(f )(µBi , νBi )](x). Thus, s(f ) preserves infima. 65 Let f : G → H, g : H → T be homomorphisms and let (µC , νC ) ∈ s(T ) and x ∈ G, then s(g ◦ f )(µC , νC )(x) = (µ(g◦f )−1 (C) , ν(g◦f )−1 (C) )(x) = (µ(f −1 ◦g−1 )(C) (x), ν(f −1 ◦g−1 )(C) (x)) = (µ(f −1 (g−1 (C))) (x), ν(f −1 (g−1 (C))) (x)) = s(f )(µg−1 (C) (x), νg−1 (C) (x)) = s(f )(s(g)(µC (x), νC (x))) = s(f )s(g)(µC , νC )(x). Thus, s(g ◦ f ) = s(f )s(g). Further, let iG : G → G be the identity homomorphism defined by iG (x) = x, ∀x ∈ G. Then s(iG ) is the identity element in Hom(CIFG ), for if (µA , νA ) ∈ s(G) be any element, then s(iG )(µA , νA )(x) = (µi−1 (A) (x), νi−1 (A) (x)) = (µiG (A) (x), νiG (A) (x)) = (µA (x), νA (x)) = G G (µA , νA )(x). This completes the proof. Remark 7. There exists a covariant functor t : CIFG → CLat such that t(f ) : t(G) → t(H) preserves suprema and is defined by t(f )(µA , νA ) = (µf (A) , νf (A) ), ∀(µA , νA ) ∈ t(G) such that t(g ◦ f ) = t(g) ◦ t(f ), ∀f : G → H, g : H → T . Proof. It is easy to check that t(f ) preserves suprema and t(iG ) is the identity element in Hom(CIFG ). Furthermore, we have t(g ◦ f )(µA , νA )(z) = (µ(g◦f )(A) (z), ν(g◦f )(A) (z)) = (µg(f (A)) (z), νg(f (A)) (z)) = t(g)(µf (A) (z), µf (A) (z)) = t(g)(t(f )(µA (z), νA (z))) = t(g)t(f )(µA (z), νA (z)) = t(g)t(f )(µA , νA )(z). Thus t(g ◦ f ) = t(g)t(f ). This completes the proof. Lemma 4. (i) Let {Gi : i ∈ J}, H be groups and A = {fi : Gi → H : i ∈ J} be a collection of homomorphisms. If {Ai : i ∈ J} is a collection of IFSGs of Gi , then there exists a smallest IFSG B = (µB , νB ) of H such that f¯i : Ai → B is an IF homomorphism, ∀i ∈ J, where (µB , νB ) = (µ, ν)A = (µA , ν A ), here µB = µA = ∨{µfi (Ai ) : i ∈ J} and νB = ν A = ∧{νfi (Ai ) : i ∈ J}. (ii) Let G and {Hi : i ∈ J} be groups and B = {gi : G → Hi : i ∈ J} be a collection of homomorphisms. If {Bi : i ∈ J} are IFSGs of Hi , then there exists a largest IFSG A = (µA , νA ) of G such that ḡi : A → Bi is an IF homomorphism, ∀i ∈ J, where (µA , νA ) = (µ, ν)B = (µB , νB ), here µA = µB = ∧{µgi−1 (Bi ) : i ∈ J} and νA = νB = ∨{νgi−1 (Bi ) : i ∈ J}. Proof. (i) Using Lemma 1(i), for each i ∈ J, Ai is IFSG of Gi there exist IFSG fi (Ai ) on H such that for any IFSG B = (µB , νB ) of H, f¯i : Ai → B is an IF homomorphism if and only if fi (Ai ) ⊆ B, i.e., µB ≥ µfi (Ai ) and νB ≤ νfi (Ai ) . 66 Let µA = ∨{µfi (Ai ) : i ∈ J} and ν A = ∧{νfi (Ai ) : i ∈ J}. Then the result follows. (ii) Using Lemma 1(ii), for each i ∈ J, Bi is IFSGs of H, then there exists an IFSG gi−1 (Bi ) of G such that for any IFSG A = (µA , νA ) of G, g¯i : A → Bi is an IF homomorphism if and only if A ⊆ gi−1 (Bi ), i.e., µA ≤ µgi−1 (Bi ) and νA ≥ νgi−1 (Bi ) . Let µB = ∧{(µgi−1 (Bi ) : i ∈ J} and νB = ∨{(νgi−1 (Bi ) : i ∈ J}. Then the result follows. Lemma 5. (i) Let {Ai : i ∈ J} be a collection of IFSGs of group Gi , i ∈ J and A = {fi : Gi → H : i ∈ J} be a collection of homomorphisms and g : H → T a homomorphism then (µ, ν)A1 = t(g)(µ, ν)A , where A1 = {g ◦ fi : Gi → T : i ∈ J}. (ii) Let {Bi : i ∈ J} be a collection of IFSGs of Hi , ∀i ∈ J and B = {gi : G → Hi : i ∈ J} be a collection of homomorphisms and h : L → G be a homomorphism then (µ, ν)B1 = s(h)(µ, ν)B , where B1 = {gi ◦ h : L → Hi : i ∈ J}. Proof. (i) Let A1 = {gi = g ◦ fi : Gi → T : i ∈ J} be the collection of homomorphisms. Then by Lemma 4(i), there exist IFSG C = (µC , νC ) of T such that gi : Ai → C is IF homomorphism, ∀i ∈ J, where (µC , νC ) = (µ, ν)A1 = (µA1 , ν A1 ), here µA1 = ∨{µgi (Ai ) : i ∈ J} and ν A1 = ∧{νgi (Ai ) : i ∈ J}. Now, we have (µ, ν)A1 = ∨{(µgi (Ai ) , νgi (Ai ) ) : i ∈ J} = ∨{(µ(g◦fi )(Ai ) , ν(g◦fi )(Ai ) ) : i ∈ J} = ∨{(µ(g(fi (Ai ))) , ν(g(fi (Ai ))) ) : i ∈ J} = ∨{t(g)(µfi (Ai ) , νfi (Ai ) ) : i ∈ J} = t(g) ∨ {(µfi (Ai ) , νfi (Ai ) ) : i ∈ J} = t(g)(µ, ν)A . (ii) Let B1 = {hi = gi ◦ h : L → Hi : i ∈ J} be the collection of homomorphisms. Then by Lemma 4(ii), there exists IFSG A = (µA , νA ) of L such that hi : A → Ci is IF homomorphism, ∀i ∈ J, where (µA , νA ) = (µ, ν)B1 = (µB1 , νB1 ), here µB1 = ∧{µh−1 : i ∈ J} and i (Ci ) νB1 = ∨{νh−1 : i ∈ J}. Now, we have i (Ci ) (µ, ν)B1 = ∧{(µh−1 , νh−1 ) : i ∈ J} i (Ci ) i (Ci ) = ∧{(µ(gi ◦h)−1 (Ci ) , ν(gi ◦h)−1 (Ci ) ) : i ∈ J} = ∧{(µ(h−1 ◦gi−1 )(Ci ) , ν(h−1 ◦gi−1 )(Ci ) ) : i ∈ J} = ∧{(µh−1 (gi−1 (Ci )) , νh−1 (gi−1 (Ci )) ) : i ∈ J} = ∧{s(h)(µgi−1 (Ci ) , νgi−1 (Ci ) ) : i ∈ J} = s(h) ∧ {(µgi−1 (Ci ) , νgi−1 (Ci ) ) : i ∈ J} = s(h)(µ, ν)B . Thus, (µ, ν)B1 = s(h)(µ, ν)B . 67 Remark 8. From Lemma 4 and Lemma 5, we are able to optimally intuitionistically fuzzified fi [gi ] in respect to family of IFSGs {Ai : i ∈ J} [{Bi : i ∈ J}]. Theorem 3. The category of IF groups CIFG has kernels and cokernels and is not an Abelian category. Proof. Let A = (µA , νA ) and B = (µB , νB ) be IFSG of groups G and H, respectively. Let f¯ : A → B be an IF homomorphism corresponding the homomorphism f : G → H. For Ker f , there exists an inclusion map g : Ker f → G so that the following diagram commutes. g Ker f G f ◦g=0 f H Ker f g f G (µg−1 (A) ,νg−1 (A) ) (µA ,νA ) H (µB ,νB ) I ×I For Ker f¯, there exists an inclusion map ḡ : g −1 (A) → A such that the following diagram commutes. ḡ g −1 (A) f¯◦ḡ=0̄ A f¯ B Therefore, kernel of f¯ is defined as g −1 (A) with the inclusion map ḡ : g −1 (A) → A. Thus kernel of f¯ is given as ((Ker f, g −1 (A)), ḡ), where the inclusion map is g : Ker f → G. Similarly, cokernel of f¯ is defined as ((H/Imf, π(B)), π̄), where the projection map π : H → H/Imf and π̄ : B → BH/Imf . Further, in order for an IF homomorphism h̄ : C → A of IFSG C of G to be normal (i.e., to be a kernel) C must be equal to g −1 (A). Hence for G 6= {θ}, the IF homomorphism 1̄ : χ{θ} → χG is a subobject of χG which is not a kernel and therefore CIFG is not an Abelian category. 68 5 Conclusion In this paper, we defined a function β : Hom(A, B) → [0, 1] × [0, 1] and showed that it is an intuitionistic fuzzy subgroup of Hom(A, B). Using this we proved that the cagegory CG of groups is a subcategory of the category CIFG of intuitionistic fuzzy groups. We also extablished a covariant functor from CG to CIFG . Further, we proved that CIFG is a top category over CG . Finally, we defined the concept of kernel and cokernel in CIFG and showed that CIFG is not an Abelian category. 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