Category of Intuitionistic Fuzzy Modules

: We study the relationship between the category of R -modules ( C R-M ) and the category of intuitionistic fuzzy modules ( C R-IFM ). We construct a category C Lat(R-IFM) of complete lattices corresponding to every object in C R-M and then show that, corresponding to each morphism in C R-M , there exists a contravariant functor from C R-IFM to the category C Lat (=union of all C Lat(R-IFM) , corresponding to each object in C R-M ) that preserve inﬁma. Then, we show that the category C R-IFM forms a top category over the category C R-M . Finally, we deﬁne and discuss the concept of kernel and cokernel in C R-IFM and show that C R-IFM is not an Abelian Category.


Introduction
The category theory is concerned with the mathematical entities and the relationships between them.Categories also emerge as unifying concepts in many fields of mathematics, particularly in all other areas of computer technology and mathematical physics.In the L.A. Zadeh [1] introductory paper, fundamental research is being carried out in the fuzzy sets context.Almost all of this mathematical development has been categorical.Several other researchers have developed and researched theories of fuzzy modules, fuzzy exact sequences of fuzzy complexes, and fuzzy homologies of fuzzy chain complexes [2][3][4][5][6].K.T. Atanassov [7,8] suggested the interpretation of intuitionistic fuzzy sets that could be a generalized form of fuzzy sets.R. Biswas was the first to apply the criterion of intuitionistic fuzzy sets in algebra and led to the introduction of an intuitionistic fuzzy subgroup of a group in [9].Later on, Hur and others in [10] and [11], brought the perception of the intuitionistic fuzzy subring and ideals.B. Davaaz and others in [12] delivered the perception of an intuitionistic fuzzy submodule of a module.Later, many mathematicians contributed to the study of intuitionistic fuzzy submodules, see [13][14][15][16][17][18][19].The focus of this study is to carry the analysis of intuitionistic fuzzy modules over a commutative ring, to a categorical approach, to pave the way for future research.
Along with the commutative ring R with unity, we defined a category (C R-IFM ) of intuitionistic fuzzy modules where the classes of all intuitionistic fuzzy modules and intuitionistic fuzzy R-homomorphisms constitute objects and morphisms.The compositions of morphisms are the ordinary compositions of functions.Moreover, we reveal that Hom(A, B) is an abelian group under the ordinary addition of R-homomorphisms, where A and B are intuitionistic fuzzy submodules.In the context of the additive composition, this structure appears to have a distributive influence on the left and at the right.This paper shows that C R-IFM seems to be an additive category, even though it is not an abelian category (Section 4).
In this approach, we are implementing an important technological tool to "optimally intuitionistic fuzzify" the R-homomorphism families.This capability to intuitionistic fuzzify provides C R-IFM with the top category structure over C R-M (Section 3).We even characterize zero objects, kernels, cokernels in C R-IFM .Our objective is to study the intuitionistic fuzzy aspects of some algebraic structures, such as rings and modules.The study of fuzzy aspects of rings and modules is well developed, even then there are many scopes for further studies in intuitionistic fuzzification of such algebraic structures.The adopted approach is better than the previously developed fuzzy approach as it includes a non-membership function, which provides a more effective and efficient tool for dealing with uncertainties.
Finally, we have shown that the category of fuzzy modules C R-FM is a subcategory of a category of intuitionistic fuzzy modules C R-IFM , and we established a contravariant functor from the category C R-IFM to the category C Lat (= union of all C Lat(R-IFM) , corresponding to each object in C R-M ).For basic definitions and results about category, we follow [20][21][22].

1.
Construct the category of intuitionistic fuzzy modules (C R-IFM .

2.
Study the relationship between the category of R-modules (C R-M ) and the category of intuitionistic fuzzy modules (C R-IFM ).

3.
Analyze the concept of kernel and cokernel in C R-IFM .

4.
Investigate that C R-IFM is not an abelian category.

Results
Throughout the paper, R is a commutative ring with unity 1 and 1 = 0. M is a unitary R-module, θ is a zero element of M, and I represents the unit interval [0, 1].

Definition 1 ([20]).
A category C is a quadruple (Ob, Hom, id, o) consisting of: (Cl) Ob, an object class; (C2) Hom(X, Y) a set of morphisms is associated with each ordered object pair (X, Y); (C3) a morphism id X ∈ Hom(X, X), for each object X; (C4) a composition law holds i.e., if f ∈ Hom(X, Y) and g ∈ Hom(Y, Z), go f ∈ Hom(X, Z); such that it satisfies the following axioms: (M3) a set of Hom(X, Y) morphisms are pairwise disjoint.

Example 1.
(1) Set, the category with sets as objects, functions as morphisms, and the usual compositions of functions, as compositions.(2) Grp, the category with groups as objects, group homomorphisms as morphisms, and their compositions as compositions.(3) Ab, the category with abelian groups as objects, group homomorphisms as morphisms, and their compositions as compositions.

Definition 2 ([21]
).The opposite category C op of the specified category C is constructed when reversing the arrows, i.e., for each ordered object pair (X, Y) There is a product and a co-product for any pair of objects of C.

3.
Each morphism in C does have a kernel and a cokernel.4.
Each monomorphism in C seems to be the kernel of its cokernel.

5.
Any epimorphism in C seems to be the cokernel of its kernel.
(iii) We call F : C → D a functor from C to D. (iv) A functor defined above is called a covariant functor that preserves:

•
The domains, the co-domains, and identities.

•
The composition of arrows, it especially retains 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 Thus, a contravariant functor F : C → D is the same as a covariant functor F : C op → D.

Definition 7 ([22]
).The category C S formed from a given category C is called a top category over C, if corresponding to every object A in C, the collection s(A) of elements of C with the ordered relation defined on it, form a complete lattice, and the inverse image map s( f ), s(B) → s(A), form a contravariant functor.
Then M is an R-module under usual componentwise addition and scalar multiplication composition.Then the intuitionistic fuzzy set A = (µ A , ν A ) of M defined by is an intuitionistic fuzzy submodule of M.
Definition 10 ( [13,19]).Let K as a submodule of an R-module M. The intuitionistic fuzzy characteristic function of K is defined by χ K , described by χ K (a) = (µ χ K (a), ν χ K (a)), where Clearly, χ K is an IFSM of M. The IFSMs χ {θ} , χ M are called trivial IFSMs of module M. Any IFSM of the module M apart from this is called proper IFSM.

Definition 11 ([17])
To avoid confusion between an R-homomorphism f : M → N and an intuitionistic fuzzy Rhomomorphism f : A → B. We denote the latter by f : A → B. So, given an IF R-homomorphism f : A → B, f : M → N is the underlying R-homomorphism of f .The set of all IF R-homs from A to B is denoted by Hom(A, B).
Example 6.Let M = ({0, 1, 2, 3, 4}, + 4 ) and N = ({0, 1}, + 2 ) be two Z-modules.Define intuitionistic fuzzy sets A = (µ A , ν A ) and B = (µ B , ν B ) on M and N, respectively, as Then A and B are intuitionistic fuzzy submodules of M and N, respectively.Define the mapping f : Proposition 2. Hom(A, B) form an additive abelian group.Moreover, it is a unitary R-module when R is a commutative ring with unity.
Similarly, we can show that ν B (( . 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: Precisely, f + 0 = 0 + f and f + − f = − f + f = 0.This shows that − f works as the additive inverse of f and 0 is the zero element (or additive identity) in Hom(A, B).Hence, Hom(A, B) is an additive abelian group.
Furthermore, we define the R-scalar multiplication on Hom(A, B) as follows: If f ∈ Hom(M, N) and f ∈ Hom(A, B), define As Ker f is the pre-image of {θ} under f , we have Ker f ⊆ Ker f .Especially, if B = χ N , then we have Ker f = A, for all f ∈ Hom(A, B).
Proposition 3. Let A and B are IFSM of R-modules M and N, respectively, and f : (ii) The restriction of A to Ker f i.e., A| Ker f is an IFSM of A. Proof.
Ker f .Now it is simple to prove that C is an IFSM of M and C ⊆ A.

Categories of Intuitionistic Fuzzy Modules
In this section, we analyze the IF-modules category and the existence of the covariant functor between the modules category and IF-modules category.
Proof.As shown in Proposition 2, Hom(A, B) is an R-module, where the scalar multiplication on Hom(A, B) is defined as (r.f )(a) = r f (a), ∀a ∈ M.
Next, we show that the function β : Hom(A, B) → I × I on R-module Hom(A, B) defined by where Further, let f , ḡ ∈ Hom(A, B) and a ∈ M. Consider Thus, .Likewise, we are able to exhibit that as defined in Theorem 1, a composition law associating to each pair of morphisms f ∈ Hom(M, N) and g ∈ Hom(N, P), a morphism go f ∈ Hom(P, Q), such that the following axioms hold: (M1) Associativity: ho(go f ) = (hog)o f , for all f ∈ Hom(M, N), g ∈ Hom(N, P) and h ∈ Hom(P, Q); (M2) preservation of morphisms: Thus, A category of IF R-modules can be constructed as Proof.It follows from Definition 3, Proposition 1 and Theorem 1.
Proposition 5.There exist a covariant functor from C R-M to C R-IFM .
We want to prove that β preserves object, composition, domain, and codomain identity.

Optimal Intuitionistic Fuzzification
In this section, we show that the category C R-IFM forms a top category over the category C R-M .To prove this, we first construct a category C Lat(R-IFM) of complete lattices corresponding to every object in C R-M and then show that corresponding to each morphism in C R-M , there exists a contravariant functor from C R-IFM to the category C Lat (=union of all C Lat(R-IFM) , corresponding to each object in C R-M ) that preserve infima.Finally, we define the notion of kernel and cokernel for the category C R-IFM and show that C R-IFM is not an abelian category.
Let A = (µ A , ν A ) and B = (µ B , ν B ) are IFSM of R-modules M and N, respectively, and f : M → N is R-homomorphism.With the help of A and f , we can provide an IF module structure on N by With the help of B and f , we can provide an IF module structure on M by Likewise, we are able to exhibit that ν In particular, we will say that for each IFSM A Proof.Let {(µ i , ν i ) : i ∈ J} be a collection of elements of s(M).Then infimum and supremum on s(M) are explicitly specified as: Then s(M) form a complete lattice.

Remark 2.
(i) The least element of s(M) is 0 and the greatest element of s(M) is 1. (ii) s(M) under the order relation defined above form a category where Ob(s(M)) = all IF-modules of M and Hom(s(M)) = order relation defined above.(iii) Supremum can also be defined as ∨ i∈J (µ i , ν i )(a) = (Sup i∈J {µ i (a)}, In f i∈J {ν i (a)}), which only holds for IF sets but does not hold for IF modules including when J is finite.
Further, i M : M → M is the identity R-homomorphism, such that i M (a) = a, ∀a ∈ M. Then s(i M ) be the identity element in Hom(C R-IFM ), for if (µ A , ν A ) ∈ s(M) be any element, then Remark 4.There exists a covariant functor t : C R-IFM → C Lat so t( f ) : t(M) → t(N) preserves suprema and is determined by t Proof.It is very simple to find that t( f ) preserves suprema and t(i M ) is the identity element in Hom(C R-IFM ).Furthermore, we have Thus t(go f ) = t(g)t( f ).Hence, the result is proved.
Lemma 4. (i) Let {M i : i ∈ J}, N are R-modules and A = { f i : M i → N : i ∈ J} be a collection of R-homomorphisms.If {A i : i ∈ J} is a collection of IFSMs of M i , then there exists a smallest IFSM B = (µ B , ν B ) of N so that fi : (ii) Let M and {N i : i ∈ J} are R-modules and B = {g i : M → N i : i ∈ J} be a collection of R-homomorphisms.If {B i : i ∈ J} are IFSMs of N i , then there exists a largest IFSM A = (µ A , ν A ) of M so that ḡi : Subsequently, the consequence follows.
(ii) Using Lemma 1(ii), for each i ∈ J, B i is IFSM of N, then there exists an IFSM g (ii) Let {B i : i ∈ J} are IFSMs of N i , ∀i ∈ J and B = {g i : M → N i : i ∈ J} be a family of R-homomorphisms and h : L → M a homomorphism then ∀i ∈ J, where (µ A , ν A ) = (µ, ν) B 1 = (µ B 1 , ν B 1 ), here µ B 1 = ∧{µ h −1 i (C i ) : i ∈ J} and ν B 1 = ∨{ν h −1 i (C i ) : i ∈ J}.Now, we have Thus, (µ, ν) g f For Ker f , there exists an inclusion map ḡ : g −1 (A) → A such that the following diagram commutes Therefore, the kernel of f is defined as g −1 (A) with the inclusion map ḡ : g −1 (A) → A. Thus, the kernel of f is given as ((ker f , g −1 (A)), ḡ), where the inclusion map is g : ker f → M.
Similarly, the cokernel of f is defined as ((N/Im f , π(B)), π), where the projection map π : N → N/Im f and π : B → B N/Im f .Remark 6.Although the category of IF modules C R-IFM has kernels and cokernels even then it is not an abelian category.
By definition of the abelian category, every monomorphism should be normal, i.e, every monomorphism is a kernel of some morphism.An IF R-hom h : C → A of IFSM C of M on being normal (i.e., being a kernel) C should be identical to g −1 (A).Consequently, for M = {θ}, the IF R-hom 1 : χ {θ} → χ M is a sub-object of χ M , which is not a kernel.Thus, C R-IFM is not an abelian category.

Discussion
In this paper, we studied the category of intuitionistic fuzzy modules C R-IFM over the category of fuzzy modules C R-M by constructing a contravariant functor from the category C R-IFM to the category C Lat (=union of all C Lat(R-IFM) , corresponding to each object in C R-M ).We showed that C R-M is a subcategory of C R-IFM .Further, we showed that C R-IFM is a top category that is not an abelian category.

Example 4 .
The category Ab is an example of an abelian category.Proposition 1([4]).The collection of all R-modules and R-homomorphisms is a category.This category is denoted by C R-M .Definition 6 ([21]).Let C = (Ob(C), Hom(C), id, o) and D = (Ob(D), Hom(D), id, o) be two categories and let F 1 : Ob(C) → Ob(D) and F 2 : Hom(C) → Hom(D) be maps.Then the quadruple F

Theorem 1 .
Let A = (µ A , ν A ) and B = (µ B , ν B ) are two IF modules of R-modules M and N respectively.Then the function β : Hom(A, B) → I × I on R-module Hom(A, B) defined by

Definition 12 .
The category C R-M = (Ob(C R-M ), Hom(C R-M ), o) has R-modules as objects and R-homomorphisms as morphisms, with composition of morphisms defined as the composition of mappings.An IF-module category C R-IFM over the base category C R-M is completely described by two mappings: α : Ob(C R-M ) → I × I; β : Hom(C R-M ) → I × I IF-module category C R-IFM consists of (C1) Ob(C R-IFM ) the set of objects as IFSMs on Ob(C R-M ), i.e., the objects of the form α : Ob(C R-M ) → I × I; (C2) Hom(C R-IFM ) the set of IF R-homomorphisms corresponding to underlying R-homomorphisms from Hom(C R-M ), i.e., IF R-homomorphisms of the form β : Hom

Remark 3 .
t(M) under the order relation defined above form a category where Ob(t(M)) = all IF-modules of M and Hom(t(M)) = order relation as defined above.Theorem 2. C R-IFM is a top category over C R-M .

Theorem 3 .
B 1 = s(h)(µ, ν) B .Remark 5. From Lemma 4 and Lemma 5, we are able to optimally intuitionistically fuzzify f i [g i ], in respect to the family of IFSMs {A i : i ∈ J} [{B i : i ∈ J}].The category of IF modules C R-IFM has kernels and cokernels.Proof.Let A = (µ A , ν A ) and B = (µ B , ν B ) be IFSM of R-modules M and N, respectively.Let f : A → B be an IF R-hom corresponding to the R-homomorphism f : M → N.For Ker f, there exists an inclusion map g : ker f → M in order for the subsequent diagram commutes (µ g −1 (A) ,ν g −1 (A) )

Example 2. The category Grp is a subcategory of Set. Definition 4 (
[21]).For the ordered object pair (X, Y) of D, a full subcategory of a category C is a category D if ob(D) ⊆ Ob(C) and Hom D (X, Y) = Hom C (X, Y).
Definition 3([21]).Category D is said to be a subcategory of the category C when ob(D) ⊆ Ob(C), Hom D (X, Y) ⊆ Hom C (X, Y) ∀ ordered object pair (X, Y) and composition of morphisms, and the identity of D should be the same as that of C.