INTUITIONISTIC H-FUZZY RELATIONS

We introduce the category IRel(H) consisting of intuitionistic fuzzy relational spaces on sets and we study structures of the category IRel(H) in the viewpoint of the topological universe introduced by Nel. Thus we show that IRel(H) satisfies all the conditions of a topological universe over Set except the terminal separator property and IRel(H) is cartesian closed over Set.


Introduction
In 1965, Zadeh [30] introduced a concept of a fuzzy set as the generalization of a crisp set.Also, in 1971, he introduced a fuzzy relation naturally, as a generalization of a crisp relation in [31].
Nel [27] introduced the notion of a topological universe which implies concrete quasitopos [1].Every topological universe satisfies all the properties of a topos except one condition on the subobject classifier.The notion of a topological universe has already been put to effective use in several areas of mathematics in [24,25,28].In 1980, Cerruti [8] introduced the category of L-fuzzy relations and investigated some of its properties.After that time, Hur [14] introduced the category Rel(H) of the fuzzy relational spaces with a complete Heyting algebra H as a codomain and he studied the category Rel(H) in the sense of a topological universe.
In 1983, Atanassov [2] introduced the concept of an intuitionistic fuzzy set as the generalization of fuzzy sets and he also investigated many properties of intuitionistic fuzzy sets (cf.[3]).After that time, Banerjee and Basnet [4], Biswas [6], and Hur and his colleagues [15,16,17,20] applied the concept of intuitionistic fuzzy sets to algebra.Also, C ¸oker [9], Hur and his colleagues [21], and S. J. Lee and E. P. Lee [26] applied one to topology.In particular, Hur and his colleagues [18] applied the notion of intuitionistic fuzzy sets to topological group.
In this paper, we introduce the category IRel(H) of intuitionistic H-fuzzy relational spaces and study the category IRel(H) in a topological universe viewpoint.In particular, we show that IRel(H) satisfies all the conditions of a topological universe over Set except the terminal separator property.Also IRel(H) is shown to be cartesian closed over Set.For general categorical background, we refer to Herrlich and Strecker [12].

Preliminaries
In this section, we will introduce some basic definitions and well-known results which are needed in the next sections.
Let X be a set, let (X i ) i∈I be a family of sets indexed by a class I, and let f i be a mapping with domain X for each i ∈ I. Then a pair (X,( f i ) I ) (simply, ( f i ) I ) is called a source of mappings.A sink of mappings is the dual notion of a source of mappings.Definition 2.1 [12].Let A be a concrete category and let I be a class.
(1) A source in A is a pair (X,( , where X is an A-object and ( f i : X → X i ) I is a family of A-morphisms each with domain X.In this case, X is called the domain of the source and the family (X i ) I is called the codomain of the source.
(2) A source (X, f i ) is called a monosource provided that the f i can be simutaneously canceled from the left; that is, provided that for any pair Y s r X of morphisms such that Dual notions: sink in A and episink.
Definition 2.2 [23].Let A be a concrete category and let ((Y i ,ξ i )) I be a family of objects in A indexed by a class I.For any set X, let ( f i : X → Y i ) I be a source of mappings indexed by I.An A-structure ξ on X is said to be initial with respect to (X,( f i ),((Y i ,ξ i ))) provided that the following conditions hold.
(2) If (Z,ρ) is an A-object and g : Z → X is mapping such that for each i ∈ Z, the mapping f i • g : (Z,ρ) → (Y i ,ξ i ) is an A-morphism, then g : (Z,ρ) → (X,ξ) is an A-morphism.In this case, ( f i : (X,ξ) → (Y i ,ξ i )) I is called an initial source in A. Dual notions: final structure and final sink.Definition 2.3 [23].A concrete category A is said to be topological over Set provided that for each set X, for any family ((Y i ,ξ i )) I of A-objects, and for any source ( f i : X → Y i ) I of mappings, there exists a unique A-structure ξ on X which is initial with respect to (X,( Dual notions: cotopological category.
Definition 2.7 [23].Let A be a concrete category.
(1) The A-fiber of a set X is the class of all A-structures on X.
(2) A is called properly fibered over Set provided that the following conditions hold.
(i) Fiber-smallness.For each set X, the A-fiber of X is a set.
(ii) Terminal separator property.For each singleton set X, the A-fiber of X has precisely one element.(iii) If ξ and η are A-structures on a set X such that 1 X : (X,ξ) → (X,η) and 1 X : (X,η) → (X,ξ) are A-morphisms, then ξ = η.
Definition 2.8 [27].A category A is called a topological universe over Set provided that the following conditions hold.
(1) A is well structured over Set, that is, (i) A is a concrete category; (ii) A has the fiber-smallness condition; (iii) A has the terminal separator property.(2) A is cotopological over Set.
(3) Final episinks in A are preserved by pullbacks, that is, for any final episink (g λ : X → Y ) Λ and any A-morphism f : W → Y , the family (e λ : U λ → W) Λ , obtained by taking the pullback of f and g λ for each λ, is again a final episink.
Definition 2.9 [29].A category A is called a topos provided that the following conditions hold.
(1) There is a terminal object U in A, that is, for each A-object A, there exists one and only one A-morphism from A to U. (2) A has equalizers, that is, for any A-objects A and B and A-morphisms there exist an A-object C and an A-morphism h : (3) A is cartesian closed; (4) there is a subobject classifier in A, that is, there is an A-object Ω and A-morphism v : U → Ω such that for each A-monomorphism m : A → A, there exists a unique A-morphism φ m : A → Ω such that the following diagram is a pullback: Remark 2.10.Let A be any category with a subobject classifier.If f is any bimorphism in A, then f is an isomorphism in A (cf. [7]).

The category IRel(H)
First we will list some concepts and one result which are needed in this section and the next section.Next, we introduce the category IRel(H) of intuitionistic H-fuzzy relational spaces and show that it has similar structures as those of ISet(H).
Definition 3.1 [5,22].A lattice H is called a complete Heyting algebra if H satisfies the following conditions: (1) H is a complete lattice; (2) for any a,b ∈ H, the set {x ∈ H : In particular, for each a ∈ H, N(a) = a → 0 is called the negation or the pseudocomplement of a.
Throughout this paper, we use H as a complete Heyting algebra.
From Definitions 3.3 and 3.4, we can form a concrete category ISet(H) consisting of all IHFSs and morphisms between them.In this case, each ISet(H)-morphism will be called an ISet(H)-mapping.[13]).Definition 3.5 [14].(1) Let X be a set.R is called an H-fuzzy relation (or simply, a fuzzy relation) on X if µ R : X × X → H is a mapping.In this case, (X,R) is called an H-fuzzy relational space (or simply, a fuzzy relational space).
(2) Let (X,R X ) and (Y ,R Y ) be any fuzzy relational spaces.A map f : From Definition 3.5, we can form a concrete category Rel(H) consisting of all relational spaces and relation preserving mappings between them.Every Rel(H)-morphism will be called a Rel (H)-mapping.
, where µ R and ν R denote the degree of membership (namely, µ R (x, y)) and the degree of nonmembership (namely, The following is the immediate result of Definition 3.7. Proposition 3.8.Let (X,R X ), (Y ,R Y ), and (Z,R Z ) be IHFRSs. (1) From Definitions 3.6 and 3.7, and Proposition 3.8, we can form a concrete category IRel(H) consisting of all IHFRSs and relation-preserving mappings between them.Every IRel(H)-morphism will be called an IRel(H)-mapping.Moreover, it is clear that if Theorem 3.9.IRel(H) is topological over Set.
Proof.Let X be any set and let ((X α ,R α )) Γ be any family of IHFRSs indexed by a class Γ.Let ( f α : X → X α ) Γ be any source of mappings.We define two mappings µ R : For any (Y ,R Y ) ∈ IRel(H), let g : Y → X be any mapping for which f α • g : (Y ,R Y ) → (X α ,R α ) is an IRel(H)-mapping for each α ∈ Γ.Then we can easily check that g : (Y ,R Y ) → (X,R) is an IRel(H)-mapping.Hence R = (µ R ,ν R ) is the initial structure on X with respect to (X,( f α ),((X α ,R α ))).This completes the proof.
Example 3.10.(1) Inverse image of an IHFR.Let X be a set, let (Y ,R Y ) be an IHFRS, and let f : X → Y be any mapping.Then there exists the initial IHFR R on X for which f : (X,R) → (Y ,R Y ) is an IRel(H)-mapping.In this case, R is called the inverse image of R Y under f .In particular, if X ⊂ Y and f : X → Y is the canonical mapping, then (X,R) is called an intuitionistic H-fuzzy relational subspace of (Y ,R Y ), where (2) Intuitionistic fuzzy product structure.Let ((X α ,R α )) Γ be any family of IHFRSs and let X = X α be the product set of (X α ) Γ .Then there exists the initial IHFR R on X for which each projection π α : (X,R) → (X α ,R α ) is an IRel(H)-mapping.In this case, R is called the product of (R α ) Γ and is denoted by Corollary 3.11.IRel(H) is complete and cocomplete.Moreover, by definition, it is easy to show that IRel(H) is well powered and co-well-powered.
From Result 2.4 and Theorem 3.9, it is clear that IRel(H) is cotopological.However, we show directly that IRel(H) is cotopological.Theorem 3.12.IRel(H) is cotopological over Set.
Proof.Let X be any set and let ((X α ,R α )) Γ be any family of IHFRSs indexed by a class Γ.Let ( f α : X α → X) Γ be any sink of mappings.We define two mappings µ R : For any (Y ,R Y ) ∈ IRel(H), let g : X → Y be any mapping for which g ) is the final structure on X with respect to (((X α ,R α )),( f α ),X).This completes the proof.Example 3.13.(1) Intuitionistic H-fuzzy quotient relation.Let (X,R) ∈ IRel(H), let ∼ be an equivalence relation on X, and let ϕ : X → X/R the canonical mapping.Then there exists the final intuitionistic H-fuzzy relation (µ X/∼ ,ν X/∼ ) on X/ ∼ for which ϕ : (X,R) → (X/ ∼, µ X/∼ ,ν X/∼ ) is an IRel(H)-mapping.In this case, (µ X/∼ ,ν X/∼ ) is called the intuitionistic H-fuzzy quotient relation of X by R.
For any singleton set {a}, since the IHFR R on {a} is not unique, the category IRel(H) is not properly fibered over Set.Hence, by Theorems 3.12 and 3.14, we obtain the following result.Proof.It is clear that IRel(H) has products by Corollary 3.11.We will show that IRel(H) has exponential objects.
For any IHFRSs X = (X,R X ) and Y = (Y ,R Y ), let Y X be the set of all mappings from X into Y .We define two mappings µ R : Then clearly (Y X ,R) ∈ IRel(H).Let Y X = (Y X ,R).Then, by the definition of R, On the other hand, Thus, by the definition of R, Hence h : Z → Y X is an IRel(H)-mapping.Moreover, h is the unique IRel(H)-mapping such that e X,Y • (1 X × h) = h.This completes the proof.

The relations between IRel(H) and Rel
Then G 1 and G 2 are functors.
) is an IRel(H)-mapping.Hence 1 X is a G 1 -universal map for (X,µ R ) in Rel(H).This completes the proof.
For each (X,µ R ) ∈ Rel(H), F 1 (X,µ R ) = (X,µ R ,N(µ R )) is called an intuitionistic Hfuzzy set in X induced by (X,µ R ).Let us denote the category of all induced intuitionistic H-fuzzy sets and IRel(H)-mappings as IRel * (H).Then it is clear IRel * (H) is a full subcategory of IRel(H).
Hence F : Rel(H) → ISet * (H) is an isomorphism.This completes the proof.Remark 4.6.We are going to investigate "intuitionistic H-fuzzy reflexive relations," "some subcategories of the category IRelk (H)," and "intuitionistic H-fuzzy relations on intuitionistic H-fuzzy sets" in the viewpoint of topological universe.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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Theorem 3 . 15 .
IRel(H) satisfies all the conditions of a topological universe over Set except the terminal separator property.Theorem 3.16.IRel(H) is cartesian closed over Set.