Abstract
Fuzzy Inf. Eng. (2012) 3: 273-291 DOI 10.1007/s12543-012-0116-y ORIGINAL ARTICLE Images and Preimages of Subobjects under the Morphisms in a New Category of Fuzzy Sets-I Aparna Jain · Naseem Ajmal Received: 8 July 2010/ Revised: 10 July 2012/ Accepted: 1 August 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract This paper is the third in a sequence of papers on categories by the same authors. In one of the papers, a new category of fuzzy sets was deﬁned and a few results were established pertaining to that special category of fuzzy sets S. Here, the concept of a fuzzy subset of a fuzzy set is deﬁned under the category S. Besides, the notions of images and preimages of fuzzy sets are also deﬁned under morphisms in the category of fuzzy sets and how smoothly these images and preimages behave under the action of these morphisms is analyzed. Finally, results have been proved on algebra of morphisms of this categoryS. Keywords Category· Fuzzy set· Subobject· Fuzzy subset· Monomorphism· Image · preimage 1. Introduction In 1967, Goguen laid a categorical foundation to the theory of fuzzy sets by introducing the category of L-fuzzy sets. Since then, several authors have studied various aspects in the categories of fuzzy sets and fuzzy groups. Most of the work done in this context is based on Goguen’s category. To have an overview of a few works done in this regard, readers are referred to [7, 14, 16-19, 23, 26]. M. Winter [23] showed how Goguen categories are a suitable extension of the theory in binary relations to the fuzzy world. F. Bayomi [7] studied the behaviour of functors to and fro between the category of crisp topological spaces and the category of L-topological spaces with a special reference to topological groups. Sergey A. Aparna Jain () Department of Mathematics, Shivaji College, University of Delhi, New Delhi, India email: jainaparna@yahoo.com Naseem Ajmal () Department of Mathematics, Zakir Hussain College, University of Delhi, New Delhi, India email: nasajmal@yahoo.com 274 Aparna Jain· Naseem Ajmal (2012) Solovyov [16, 17] introduced a category X(A), which was a generalization of the category of lattice valued subsets of A. Dan Ralescu [14], in his paper, deﬁned category of C-sets, again a generalization of Goguen’s category of L-fuzzy sets. He replaced a lattice L by an arbitrary category C. Then the degree of membership is no longer a point in a lattice. It is rather an object in a category. Various points in that category of C-sets can thus be compared using morphisms between their membership degrees. Lawrence N. Stout [18, 19], tried to compare fuzzy logic and topos logic in which basis was again Goguen’s category. Zaidi and Ansari [26] introduced a few subcategories of Goguen’s category of L-fuzzy subgroups and studied their properties. It is worthwhile here to mention one of the statements quoted in the works of Lawrence N. Stout “In Goguen’s category, objects have fuzzy boundary with fuzziness measured in L, but the maps are crisp. Goguen suggests that a nicer category may result in taking maps which are fuzzy as well.” In [4], A. Jain and N. Ajmal introduced a new categoryG of fuzzy groups in which the object class was the class of all fuzzy groups in all groups. This category differed from the already known Goguen’s category [9] of fuzzy groups in the sense that the two categories comprised of different notions of morphisms. Thus, whereas most of the work done by various authors was based on Goguen’s category with the nature of object changed, here we propose a category in which morphisms are changed. Ever since the introduction of Metatheorem and subdirect product theorem by Tom Head [20] and the work of A. Weinberger [21, 22], it is sufﬁciently clear that the concept of a fuzzy group comprises of simple ﬁbring of groups. Motivated by the works of Tom Head and Weinberger, these authors felt that there was a need to deﬁne the notion of a morphism in the category G of fuzzy groups in such a way that they consisted of a ﬁbring of mappings (homomorphisms), which acted differently on different ﬁbres, and the same was achieved in our paper [5]. Moreover, it was due to the nature of these morphisms that we were able to construct the reﬂective subcategories in our category of fuzzy groups in [4]. It is to be noticed that such type of subcategories do not exist in the Goguen’s framework. Dan Ralescu in [14] deﬁned fuzzy subobjects of usual categories. On the other hand, in our category of fuzzy groups discussed in [5], objects were structured fuzzy sets and we considered the categorical subobjects of these objects. In Goguen’s category of fuzzy groups, a morphism between two objects (X,μ) and (Y,η) was simply a group homomorphism f between their underlying groups satisfying the property thatμ(x) ≤ η ( f (x)),∀ x ∈ X. Whereas in our category of fuzzy groupsG, a morphism between twoG-objects is a family of homomorphisms between their level subgroups satisfying a few obvious chain conditions. In [4], the nature of morphisms in the categoryG was discussed. Consequently, the important notion of a fuzzy subgroup of a fuzzy group was introduced in [5]. It was also demonstrated in [4] that this categoryG has uncountabely many reﬂective subcategories. In [5], the category G of fuzzy groups was further studied and parallel category S of fuzzy sets was deﬁned, whose object class is the class of all fuzzy sets in all sets and a morphism between two S-objects is not just a single mapping between their parent sets, rather it is a family of mappings between their level subsets, again with a few obvious chain conditions. In [4], the subobjects of the category G were Fuzzy Inf. Eng. (2012) 3: 273-291 275 considered and the characterization of monomorphisms gives rise to the notion of a fuzzy subgroup of a fuzzy group in the same way as the notion of subgroup of a group arises in the category Grp of ordinary groups. In [5] the notion of fuzzy subset of a fuzzy set in the categoryS was deﬁned. The notions of lower well ordered and upper well ordered fuzzy subsets were also deﬁned and using these notions, it was proved that S(μ), the collection of all fuzzy subsets of a fuzzy set μ is a complete lattice if μ satisﬁes the property of being lower or upper well ordered. A similar result was proved for L(μ), the collection of all fuzzy subgroups of a fuzzy groupμ. In the present paper, we examine the category S in the framework of algebra of morphisms. The images and preimages of a fuzzy subset of a fuzzy set under an S-morphism are deﬁned. These deﬁnitions are carefully formulated using the chain properties of level subsets of a fuzzy set. Following this, we prove some interesting results on algebra of morphisms in the category S. Then, a subcategory S of the category S is deﬁned with a restricted object class as compared to the object class of S. The object class of S consists of all fuzzy sets with ﬁnite range sets, and its morphism class is the same as that of S. Some very important results on algebra of morphisms can be obtained in the categoryS (see Propositions 4.1, 4.2, 4.4 and 4.5). All these results show the compatibility of our deﬁnitions ofS, images and preimages of fuzzy subsets with that of algebraic properties of sets that exist in classical set theory. 2. Preliminaries Zadeh [25] deﬁned a fuzzy set as a function from a nonempty set to a closed unit interval. Deﬁnition 2.1 [8] Let μ be a fuzzy set in a set X and let t ∈ [0, 1]. Then the t-cut μ ofμ is deﬁned as: μ = {x ∈ X/μ(x) ≥ t}. Observe that if t > s, then μ ⊆ μ . t s Assume that the reader is familiar with the deﬁnition of a category, a few related concepts of category theory are recalled in the following deﬁnitions. Here C is any category and A, B areC-objects. Deﬁnition 2.2 [10] A C-morphism f : A → B is said to be a monomorphism in C if for all C-morphisms h and k such that f ◦ h = f ◦ k, it follows that h = k. Deﬁnition 2.3 [10] Let A, Bbe C-objects. If f : A → B is a monomorphism. Then (A, f ) is called a subobject of B. Deﬁnition 2.4 [10] A categoryF is a subcategory of a categoryH.If (i) Ob(F ) ⊆ Ob(H), (ii) [A, B] ⊆ [A, B] ∀ A, B ∈ Ob(F ), where [A, B] . denotes the collection of all F H F F -morphisms from A to B. 276 Aparna Jain· Naseem Ajmal (2012) (iii) EveryF identity is an H identity. (iv) Composition function of F is the restriction of the corresponding function of H. If in additionF satisﬁes the condition: (v) [A, B] = [A, B] ∀ A, B ∈ Ob(F ), thenF is called a full subcategory of H. F H Readers are referred to [1-3, 6, 15, 24] for details on fuzzy sets and categories. 3. Category S of Fuzzy Sets and Image, Preimage of a Fuzzy Subset under an S-morphism We ﬁrst recall the deﬁnition of the CategoryS of fuzzy sets from [5]. Notice that the object class of S consists of ordered pairs (X,μ), where X is a set and μ is a fuzzy subset of X. We shall call the pair (X,μ) a fuzzy set in the category S. When there is no likelihood of any confusion about the base set, we shall brieﬂy denote it by μ. A similar phrase “A fuzzy group in a group G” used instead of saying “A fuzzy subgroup in a group G” is used for the objects in the categoryG of fuzzy groups. This is because the objects in our category G are fuzzy groups in all groups and a fuzzy subgroup of a fuzzy group is deﬁned using the notion of a subobject in the category G of fuzzy groups and similar notions and phrases are used in the categoryS. Deﬁnition 3.1 S is the quintupleS=(O,M,dom,cod,o), where (i) O is the class of all fuzzy sets in all sets. Members of O are called S-objects. (ii) M is the class of allS-morphisms, where anS-morphism is a relation f between twoS-objects μ andθ deﬁned as follows: f : μ → θ, f is a pair f = ({ f } ,α), t t∈Imμ where the following axioms are satisﬁed: (a) α :Imμ → Imθ is an order preserving map. (b) ∀ t ∈ Imμ, f : μ → θ is a mapping. t t α(t) (c) If t > t in Imμ, A and B are subsets of μ and μ respectively, such that i j t t i j A ⊆ B, then f (A) ⊆ f (B). t t i j (d) If t > t in Imμ, C and D are subsets of θ and θ respectively such i j α(t ) α(t ) i j −1 −1 that C ⊆ D, then f (C) ⊆ f (D). t t i j (iii) Dom and cod are functions from M to O. If f is a morphism from μ to θ, then dom ( f ) = μ and cod ( f ) = θ. (iv) ‘O’ is a function from D = {( f, g)/ f, g ∈ M and dom ( f ) = cod(g)} into M, called the composition law of S. Let ( f, g) ∈ D,μ =dom (g),η =cod ( f ) and dom ( f ) = cod (g) = θ such that f = ({ f } ,α), g = ({g } ). Deﬁne the r r∈Imθ t t∈Imμ,β composition of f and g as f ◦ g = ({ f ◦ g } ,α,◦β). Since f ◦ g turns out β(t) t t∈Imμ to be an S-morphism, therefore we set O( f, g) = f ◦ g. Fuzzy Inf. Eng. (2012) 3: 273-291 277 It can be easily veriﬁed that f ◦ g is in fact anS-morphism. Moreover, the identity morphisms exist in S and the composition of morphisms satisfy associativity. Let us brieﬂy recall the deﬁnition of the category G of fuzzy groups introduced in [4]. The objects of the category G are ordered pairs (G,μ), where G is a group and μ is a fuzzy subgroup of G. The pair (G,μ) is called a fuzzy group and brieﬂy denoted by μ. The morphism class of G consists of pairs f = ({ f } ,α), where f is a t t∈Imμ morphism from the fuzzy groupμ to the fuzzy groupθ. Hereα is an order preserving map from Imμ to Imθ and { f } is a family of homomorphisms from the level t t∈Imμ subgroup μ to the level subgroup θ satisfying the other axioms of Deﬁnition 3.1. t α(t) Readers are referred to [4] for details. Let us recall here the following notion from [4] which gives rise to the concept of a fuzzy subgroup of a fuzzy group in the categoryG. Deﬁnition 3.2 A G-morphism f = ({ f } ,α) is said to be an M-morphism if α is t t∈Imμ injective and each f for t ∈ Imμ is also injective. Following that is the characterisation of the monomorphisms ofG. Theorem 3.1 [4] A G-morphism f is an M-morphism if and only if f is a monomorphism. It is clear by Deﬁnition 2.3 that in any category, a subobject of an object is a pair consisting of an object and a monomorphism. Most often, a subobject is identiﬁed by its associated monomorphism. Moreover, it is well known that in any category whose objects are algebraic structures, subobjects give rise to subalgebras. For example, in the category Grp of ordinary groups, subobjects of objects arising from monomorphisms give rise to the notion of subgroup of a group. Further, in any category of algebraic structures in which the images and preimages of objects under morphisms are deﬁned, the notion of a subalgebra arises naturally. For example, if (G, f ) is a subobject in the category Grp, then f (G) is a subgroup of its codomain. In the reverse direction, if there is a subgroup G of a group H, then the pair (G, i) provides a subobject where i is the inclusion map from G to H. Thus, there is a one to one correspondence between the subobjects of an object (group) in the category Grp and the subgroups of that group. In our category G of fuzzy groups, since the images of objects under morphisms are deﬁned [5], a similar treatment is carried out to formulate the notion of a fuzzy subgroup of a fuzzy group. We discuss here the motivation behind the three axioms in Deﬁnition 3.3. Let ((G,μ), f ) be a subobject in the category G and let (H,θ)be the codomain of f . Then f is the pair f = ({ f } ,α), where α is an injective t t∈Imμ order preserving map from Imμ to Imθ and for each t ∈ Imμ, f is an injective homomorphism fromμ toθ (in view of Theorem 3.1). t α(t) Notice that for each t ∈ Imμ, f is a group homomorphism from the subgroupμ to t t the subgroup θ in the following result. α(t) Proposition 3.1 Let (X,μ), (Y,θ) be G-objects, f :(X,μ) → (Y,θ) be a G-morphism, f = ({ f } ,α) and Imμ = {t} . Then t t∈Imμ i i∈Λ f (μ) = f (μ )∀ α(t ) ∈ Im f (μ). α(t ) t t i i i i 278 Aparna Jain· Naseem Ajmal (2012) Now, f (μ ) is a subgroup of θ . Further, due to Axiom 3 of a G-morphism, t t α(t) { f (μ )} is an ascending chain of subgroups of H and thus the union f (μ )is t t t∈Imμ t t t∈Imμ a subgroup of H. Therefore, we have the following: (i) f (μ ) is a subgroup of H, t t t∈Imμ (ii) α(Imμ) ⊆ Imθ, (iii) f (μ) is a subgroup ofθ ∀α(t) ∈ Im f (μ). α(t) α(t) Thus, any subobject in the category G gives rise to these three properties. These facts motivated us to deﬁne a fuzzy subgroup of a fuzzy group in G in [5] given here by Deﬁnition 3.3. Notice that in the following deﬁnition, since μ and θ are fuzzy groups in G and H respectively, the level subsets μ and θ are subgroups of G and H t t respectively. Deﬁnition 3.3 [5] A fuzzy group (G,μ) is said to be a fuzzy subgroup of a fuzzy group (H,θ) if (i) G is a subgroup of H, (ii) Imμ ⊆ Imθ, (iii) μ is a subgroup of θ ∀ t Imμ. t t Now, in the reverse direction, let (G,μ), (H,θ)be G-objects satisfying the above three conditions, i.e., G is a subgroup of H,Imμ ⊆ Imθ, and μ is a subgroup of θ ∀ t ∈ Imμ. Then, we can deﬁne a monomorphism I = ({I } , i), where t t t∈Imμ i :Imμ → Imθ is the inclusion map and for each t ∈ Imμ, I : μ → θ = θ t t i(t) t is the inclusion homomorphism, thus providing us with a subobject ((G,μ), I) in the category G. Notice that I (μ ) = G. t t t∈Imμ We now deﬁne the notion of a fuzzy subset of a fuzzy set in the categoryS: Deﬁnition 3.4 A fuzzy set (X ,μ ) is said to be a fuzzy subset of a fuzzy set (X,μ) if (i) X ⊆ X, (ii) Imμ ⊆ Imμ, (iii) μ ⊆ μ ∀ t ∈ Imμ . t t A fuzzy set μ which is a fuzzy subset of a fuzzy set μ, will be denoted by μ μ or (X ,μ ) (X,μ). Notice that the third axiom in the above deﬁnition is equivalent to saying that μ (x) ≤ μ(x) ∀ x ∈ X . We now introduce the concepts of image and preimage of a fuzzy subset of a fuzzy set under an S-morphism f . Deﬁnition 3.5 Let (X,μ), (X ,μ ) and (Y,η) be S-objects such that (X ,μ ) (X,μ). Let f :(X,μ) → (Y,η) be an S-morphism, f = ({ f } ,α). Let Imμ = {t} .We t t∈Imμ i i∈Λ Fuzzy Inf. Eng. (2012) 3: 273-291 279 deﬁne f (μ ), the image of the fuzzy subset (X ,μ ) of (X,μ) under the morphism f as a fuzzy set in the union of the family{ f (μ )} as follows: t t t∈imμ f (μ ): f (μ ) → [0, 1], t∈Imμ f (μ )(y) = α(t ) if y ∈ f (μ )− f (μ ), where t, t ∈ Im(μ ). i t t i j i t j t i j t >t j i Note that ( f (μ ), f (μ )) is an S-object. t∈Imμ Deﬁnition 3.6 Let (X,μ), (Y,η) and (Y ,η ) be S-objects such that (Y ,η ) (Y,η). Let f :(X,μ) → (Y,η) be an S-morphism, f = ({ f } ,α) and Imη = {p } .We t t∈Imμ k k∈Ω −1 deﬁne f (η ), the preimage of the fuzzy subset (Y ,η ) of (Y,η) under the morphism −1 −1 f as a fuzzy set in the union of the family { f (η )}, where p ∈ Imη , t ∈ α (p ) k k k t p k k as follows: −1 −1 f (η ): f (η ) → [0, 1], t p k k −1 t ∈α (p ) k k p ∈Imη −1 −1 −1 f (η )(x) = t if x ∈ f (η )− f (η ). t p t p k k j j t >t j k −1 t ∈α (p ) j j p ≥p in Imη j k −1 −1 Here too, note that f (η ), f (η ) ∈ Ob(S). t p k k −1 t ∈α (p ) k k p ∈Imη Lemma 3.1 Let (X,μ), (X ,μ ) and (Y,η) be S-objects such that (X ,μ ) (X,μ), f :(X,μ) → (Y,η) be an S-morphism, f = ({ f } ,α) and Imμ = {t} . Then t t∈Imμ i i∈Λ f (μ ) = f (μ )∀ α(t ) ∈ Im f (μ ). α(t ) t t i i i i Proof Let α(t ) ∈ Im f (μ ) and y ∈ f (μ ). Suppose if possible, i t i t f (μ )(y)<α(t ). That is α(t )<α(t ), where f (μ )(y) = α(t ). k i k This implies t < t . Now, since f (μ )(y) = α(t ), therefore k i k y ∈ f (μ ) and y f (μ )∀ t > t in Imμ . t t n k k t n t k n This is a contradiction to the fact that t > t in Imμ , and y ∈ f (μ ). Hence i k t i t f (μ )(y) ≥ α(t ). That is y ∈ f (μ ) and thus α(t ) f (μ ) ⊆ f (μ ) . t t α(t ) i i i 280 Aparna Jain· Naseem Ajmal (2012) To show the reverse inclusion, let y ∈ f (μ ) . Then α(t ) f (μ )(y) ≥ α(t ). That is α(t ) ≥ α(t ), where f (μ )(y) = α(t ). j i j Case I: α(t ) = α(t ). Then f (μ )(y) = α(t ) and therefore by Deﬁnition 3.5, j i i y ∈ f (μ ). i t Case II: α(t ) α(t ). Then α(t ) >α(t ). Since α is an order preserving map, this j i j i implies t > t. j i Now t , t ∈ Imμ . Therefore, j i μ μ . t t j i Then by Axiom (iii) of an S-morphism, we have f (μ ) ⊆ f (μ t ). t t i j t i Since f (μ )(y) = α(t ), we have by Deﬁnition 3.5, y ∈ f (μ ). Hence y ∈ f (μ ). j t t j t i t j i Tus f (μ ) ⊆ f (μ ). α(t ) t i i t This gives the required equality. Proposition 3.2 Let (X,μ), (X ,μ ) and (Y,η) beS-objects such that (X ,μ ) (X,μ), f :(X,μ) → (Y,η) be an S-morphism, f = ({ f } ,α) and Imμ = {t} . Then t t∈Imμ i i∈Λ f (μ ) is a fuzzy subset ofη, that is f (μ ), f (μ ) (y,η). t∈Imμ Proof Since μ μ,wehaveImμ Imμ andμ ⊆ μ ∀ t ∈ Imμ . This implies f (μ ) ⊆ η ⊆ Y ∀ t ∈ Imμ . t α(t) Thus f (μ ) ⊆ Y. t∈Imμ Next, let α(t ) ∈ Im f (μ ). By Deﬁnition 3.5, this implies t ∈ Imμ ⊆ Imμ. i i Thereforeα(t ) ∈ Imη. Thus Im f (μ ) ⊆ Imη. Finally, to show that f (μ ) ⊆ η ∀ α(t ) ∈ Im f (μ ), let α(t ) ∈ Im f (μ ). α(t ) α(t ) i i i i Then t ∈ Imμ ⊆ Imμ. Since f is a map from μ to η ,we have f (μ ) ⊆ η . i t t α(t ) t α(t ) i i i i t i Therefore, in view of Lemma 3.1, we have f (μ ) ⊆ η ∀ α(t ) ∈ Im f (μ ). α(t ) α t ) i i ( i Hence f (μ ) is a fuzzy subset of η. Fuzzy Inf. Eng. (2012) 3: 273-291 281 Lemma 3.2 Let (X,μ), (Y ,η ) and (Y,η) be S-objects such that (Y ,η ) (Y,η) and f :(X,μ) → (Y,η) be an S-morphism. Then for p ∈ Imη such that α(t ) = p , k k k −1 −1 f (η ) = f (η ). k t p k k −1 Proof Let p ∈ Imη such that α(t ) = p for some t ∈ Imμ. Let x ∈ f (η ) . k k k k t Then −1 f (η )(x) ≥ t . −1 −1 Now, if f (η )(x) = t , then we have t ≥ t . By Deﬁnition 3.6, f (η )(x) = t j j k j implies −1 x ∈ f (η ). t p j j −1 If t = t , then clearly x ∈ f (η ). And if t > t , then by Axiom 4 of an S- j k j k t p k k −1 −1 −1 morphism f (η ) ⊆ f (η ). This implies x ∈ f (η ). Thus we have in both the t p t p t p j j k k k k cases, −1 x ∈ f (η ). t p k k That is −1 −1 f (η ) ⊆ f (η ). k t p k k −1 −1 To prove the reverse inclusion, let x ∈ f (η ). Suppose, if possible f (η )(x) < t p k k −1 −1 t .If f (η )(x) = t , then we have t < t . Also, by Deﬁnition 3.6, f (η )(x) = t k i i k i implies −1 −1 x ∈ f (η ) and x f (η ), ∀ t > t, p ≥ p in Imη,α(t ) = p . n i n i n n t p t p i i n n −1 Thus, keeping into consideration t > t , we get x f (η ). This contradiction k i t p k k −1 −1 establishes that f (η )(x) ≥ t . Hence x ∈ f (η ) . Therefore k t −1 −1 f (η ) ⊆ f (η ) . t p t k k k We thus get the required equality. Proposition 3.3 If (X,μ), (Y ,η ) and (Y,η) be S-objects such that (Y ,η ) (Y,η) −1 and f :(X,μ) → (Y,η) be anS-morphism, then f (η ) is a fuzzy subset ofμ. That is, −1 −1 f (η ), f (η ) (X,μ). t p k k −1 t ∈α (p ) k k p ∈Imη Proof It is easy to verify that −1 f (η ) ⊆ X. (1) t p k k −1 t ∈α (p ) k k p ∈Imη k 282 Aparna Jain· Naseem Ajmal (2012) Since η η,we have Imη ⊆ Imη and α is a map from Imμ to Imη. Therefore, ∀ −1 p ∈ Imη ,α (p ) ⊆ Imμ. Thus, in view of Deﬁnition 3.6, k k −1 Im f (η ) ⊆ Imμ. (2) −1 −1 −1 Finally to show that f (η ) ⊆ μ ∀ t ∈ Im f (η ), let t ∈ Im f (η ). Then, by t t Lemma 3.2, −1 −1 f (η ) = f (η ). t α(t) Since f is a map fromμ toη andη ⊆ η ∀ α(t) ∈ Imη ,wehave t t α(t) α(t) α(t) −1 f (η ) ⊆ μ . α(t) Hence −1 −1 f (η ) ⊆ μ ∀ t ∈ Im f (η ). (3) t t −1 By (1), (2) and (3), we get that f (η ) is a fuzzy subset ofμ. Following lemmas are easy to verify: Lemma 3.3 Let (X,μ), (Y,η), (X ,μ ) ∈ Ob(S),f :(X,μ) → (Y,η) be anS-morphism and (X ,μ ) (X,μ).If t < t in Imμ , then f (μ ) f (μ ). i j t t j t i t j i Lemma 3.4 Let (X,μ), (Y ,η ) and (Y,η) ∈ Ob(S),f :(X,μ) → (Y,η) be an S- morphism and (Y ,η ) (Y,η) such that f s are surjective ∀ t ∈ Imμ. Then for any p < p in Imη , we have i j −1 −1 f (η ) f (η ), where α(t ) = p andα(t ) = p . i i j j t p t p j j i i 4. A Subcategory S ofS and Algebra of Morphisms in the CategoryS f f We shall now restrict the class of objects in the category S and hence construct a subcategory S of S. The object class of S consists of all fuzzy sets with ﬁnite f f range sets and the morphisms. Considered in S are the same as in S. One can observe thatS is a full subcategory ofS. Some very important results on algebra of morphisms are achievable for this subcategory. Proposition 4.1 If (X,μ), (X ,μ ), (Y,η) ∈ Ob(S ), such that (X ,μ ) (X,μ) and f :(X,μ) → (Y,η) is anS -morphism, then α :Imμ → Im f (μ ) is a bijection. Proof We ﬁrst prove that ∀ t ∈ Imμ ,α(t ) ∈ Im f (μ ). For this let t ∈ Imμ . Then i i i t = μ (x) for some x ∈ X . This implies x ∈ μ . Since f is a map from μ to η i t t α(t ) t i i i andμ ⊆ μ , therefore we have t i f (x) ∈ f (μ ). t t i i t Thus f (μ ) φ. i t Case I: t = sup Imμ . Now since f (μ ) φ, let y ∈ f (μ ). Then by Deﬁnition 3.5, i t t i t i t i i f (μ )(y) = α(t ). That is,α(t ) ∈ Im f (μ ). i i Fuzzy Inf. Eng. (2012) 3: 273-291 283 Case II: t < sup Imμ . Then t < t for some t ∈ Imμ . This implies i i j j μ μ . t t j i Then by Lemma 3.3 f (μ ) f (μ ). t t j t i t j i This is true∀ t > t in Imμ . Therefore j i f (μ )− f (μ ) φ. t t i t j t i j t >t j i t ,t ∈Imμ i j Thus by Deﬁnition 3.5, α(t ) ∈ Im f (μ ). Now to prove thatα is injective, let t, t ∈ Imμ such thatα(t ) = α(t )inIm f (μ ). i j i j Setting α(t ) = α(t ) = p, suppose if possible t > t in Imμ ⊆ Imμ. Then by i j i j Lemma 3.3, f (μ ) f (μ ). t t i t j t i j This by Lemma 3.1 implies f (μ ) f (μ ) . α ti) α(t ) ( j That is f (μ ) f (μ ) . p p This contradiction implies t ≤ t . Similarly, we shall get t ≤ t . Thus t = t which i j j i i j proves thatα is injective. Also, by Deﬁnition 3.5, it is clear that if p ∈ Im f (μ ), then p = α(t ) for some t ∈ Imμ . Thus we get that α is surjective. k k k Proposition 4.2 Let (X,μ), (Y,η) ∈ Ob(S ) and f :(X,μ) → (Y,η) be an S - f f morphism. Then (X ,μ ) (X ,μ ) (X,μ) imply that f (μ ) f (μ ). 1 1 2 2 1 2 Proof It is easy to verify that f (μ ) ⊆ f (μ ). t 1 t 2 t t t∈Imμ t∈Imμ 1 2 Now to prove that Im f (μ ) ⊆ Im f (μ ), let α(t ) ∈ Im f (μ ). Then 1 2 i 1 t ∈ Imμ ⊆ Imμ . i 1 2 Case I: t = sup Imμ .We have α(t ) ∈ Im f (μ ). This implies ∃ y ∈ f (μ ), i 2 i 1 t 1 t∈Imμ such that f (μ )(y) = α(t ). Then by Deﬁnition 3.5, we have y ∈ f (μ ) ⊆ f (μ ). 1 i t 1 t 2 i t i t i i This implies f (μ ) ∅. Again by Deﬁnition 3.5, f (μ )(y) = α(t ). That is t 2 2 i i t α(t ) ∈ Im f (μ ). i 2 284 Aparna Jain· Naseem Ajmal (2012) Case II: t < sup Imμ . Let t ∈ Imμ such that t > t . Then i 2 j 2 j i μ μ . 2 2 t t j i This by Lemma 3.3 implies f (μ ) f (μ ). t 2 t 2 j t i ti Now since μ has ﬁnite range set, f (μ )− f (μ ) ∅. t 2 t 2 i t j t i j t >t j i t ,t ∈Imμ i j 2 By Deﬁnition 3.5, this implies α(t ) ∈ Im f (μ ). i 2 Hence Im f (μ ) ⊆ Im f (μ ). 1 2 Finally, we show that f (μ ) ⊆ f (μ ) ∀ α(t ) ∈ Im f (μ ). 1 α(t ) 2 a(t ) i 1 i i Let α(t ) ∈ Im f (μ ) and y ∈ f (μ ) . Then f (μ )(y) ≥ α(t ). Suppose, if possible i 1 1 a(t ) 1 i f (μ )(y)<α(t ). Then 2 i f (μ )(y)<α(t ) ≤ f (μ )(y). 2 i 1 Setting f (μ )(y) = α(t ) and f (μ )(y) = α(t ), we get that 2 k 1 j α(t )<α(t ). k j Since α is order preserving, t < t . k j Now since f (μ )(y) = α(t ), we have t ∈ Imμ .Asμ μ , we get 1 j j 1 1 2 μ ⊆ μ . 1 2 t t j j This implies f (μ ) ⊆ f (μ ). t 1 t 2 j t j t j j Again, f (μ )(y) = α(t ) by Deﬁnition 3.5 implies that y ∈ f (μ ). Thus, 1 j t 1 j t y ∈ f (μ ). (4) t 2 j t j Fuzzy Inf. Eng. (2012) 3: 273-291 285 Since f (μ )(y) = α(t ), again by Deﬁnition 3.5 we get y ∈ f (μ ) and y f (μ ) ∀ 2 k t 2 t 2 k t n t t > t in Imμ . Therefore y f (μ ) which contradicts (4). Hence n k 2 t 2 j t f (μ )(y) ≥ α(t ). 2 i This implies y ∈ f (μ ) . 2 α(t ) Thus f (μ ) ⊆ f (μ ) , ∀ α(t ) ∈ Im f (μ ). 1 α(t ) 2 α(t ) i 1 i i Hence f (μ ) f (μ ). 1 2 In the categoryS of fuzzy sets, we have the following result, the proof being similar to that of Proposition 4.2 is omitted. Proposition 4.3 Let (X,μ), (Y,η) ∈ Ob(S) and f :(X,μ) → (Y,η) be anS-morphism. Then (X ,μ ) (X ,μ ) (X,μ) imply the following: 1 1 2 2 (i) f (μ ) ⊆ f (μ ). t 1 t 2 t t t∈Imμ t∈Imμ 1 2 (ii) f (μ ) ⊆ f (μ ) ∀α(t ) ∈ Im f (μ ). 1 α(t ) 2 α(t ) i 1 i i Proposition 4.4 Let (X,μ), (X ,μ ), (Y,η) ∈ Ob(S ) such that (X ,μ ) (X,μ) and −1 f :(X,μ) → (Y,η) be an S -morphism. Then μ f ( f (μ )). Proof First recall by Proposition 3.2, f (μ ) is a fuzzy subset of η. For the sake of convenience, denote f (μ )by η . It can be easily veriﬁed that −1 X ⊆ f (η ). t p k k −1 t ∈α (p ) k k p ∈Imη −1 Now we show that Imμ ⊆ Im f (η ). Let t ∈ Imμ . Case I: t = sup Imμ . By Proposition 4.1, α :Imμ → Imη is a bijection and is order preserving, therefore p = α(t ) = sup Imη . Since t ∈ Imμ , t = μ (x) for k k k k some x ∈ X . Thus x ∈ μ . Therefore f (x) ∈ f (μ ) = f (μ ) (by Lemma 3.1) t t α(t ) k k t k = η . −1 −1 Thus x ∈ f (η ). Since p = sup Imη , f (η )(x) = t by Deﬁnition 3.6. Therefore k k t p k k −1 t ∈ Im f (η ). Case II: t < sup Imμ . Then by Proposition 4.1, p = α(t ) < sup Imη . Let k k k p ∈ Imη such that p < p and letα(t ) = p . Then we have t < t in Imμ ⊆ Imμ. j k j j j k j Therefore by Axiom 4 of anS-morphism, −1 −1 f (η ) ⊆ f (η ). t p t p j j k k 286 Aparna Jain· Naseem Ajmal (2012) Since p > p in Imη ,η η . Let y ∈ η such that y η . By Lemma 3.1 j k p p p p j k k j y ∈ η = f (μ ) = f (μ ). p α(t ) t t k k k k This implies y = f (x) for some x ∈ μ . k k Therefore −1 −1 x ∈ f (y) ⊆ f (η ). t t p k k k −1 Suppose, if possible x ∈ f (η ). Since t > t in Imμ, by Axiom 4 of an j k t p j j S-morphism, we have −1 −1 f (η ) ⊆ f (η ). t p t p j j k j Thus −1 x ∈ f (η ). t p k j This implies y = f (x) ∈ η . k p This contradiction implies −1 x f (η ). t p j j Hence −1 −1 f (η ) f (η ) ∀ p > p in Imη . j k t p t p j j k k Therefore −1 −1 f (η )− f (η ) φ. t p t p k k j j t >t j k −1 t ∈α (p ) j j p >p in Imη j k By Deﬁnition 3.6, this implies −1 t ∈ Im f (η ). We thus have −1 Imμ ⊆ Im f (η ). −1 Finally, we show that μ ⊆ f (η ) ∀ t ∈ Imμ . For this let t ∈ Imμ and let t k k t k −1 −1 x ∈ μ . Then μ (x) ≥ t . Suppose, if possible f (η )(x) < t . Let f (η )(x) = t . k k i k Fuzzy Inf. Eng. (2012) 3: 273-291 287 Then t < t in Imμ implies p < p in Imη , where α(t ) = p . Also in view of i k i k i i −1 Deﬁnition 3.6 and Proposition 4.1, f (η )(x) = t implies −1 −1 x ∈ f (η ) and x f (η ), t p t p i i n n ∀ p > p in Imη , t > t, such thatα(t ) = p . (5) n i n i n n Now, x ∈ μ implies f (x) ∈ f (μ ). t t t k k k By Lemma 3.1 we have f (μ ) = f (μ ) = η . t α(t ) k t k p k k Thus −1 x ∈ f (η ), where p > p in Imη . k i t p k k This contradicts (5). Hence we have −1 f (η )(x) ≥ t . −1 That is x ∈ f (η ) and thus −1 μ ⊆ f (η ) ∀ t ∈ Imμ . t k t k Hence we arrive at the required conclusion. That is −1 μ f ( f (μ )). Proposition 4.5 Let (X,μ), (Y,η), (Y ,η ), (Y ,η ) ∈ Ob(S ), f :(X,μ) → (Y,η) be 1 1 2 2 f an S -morphism, f = ({ f } ,α) such that f ’s are surjective ∀ t ∈ Imμ. Then f t t∈Imμ t −1 −1 (Y ,η ) (Y ,η ) (Y,η) imply that f (η ) f (η ). 1 1 2 2 1 2 Proof It is easy to verify that −1 −1 f (η ) ⊆ f (η ). t 1p t 2p k k k k p ∈Imη p ∈Imη k 1 k 2 α(t )=p α(t )=p k k k k −1 −1 −1 Next we prove that Im f (η ) ⊆ Im f (η ). Let t ∈ Im f (η ). Then by 1 2 k 1 Deﬁnition 3.6, we have α(t ) = p ∈ Imη . Since η η η, Imη ⊆ Imη . k k 1 1 2 1 2 Thus p ∈ Imη . k 2 Case I: p = sup Imη . Since p ∈ Imη , ∃ y ∈ Y such that η (y) = p . That is k 2 k 2 2 2 k y ∈ η . Since f : μ → η is a surjective map and η ⊆ η , ∃ x ∈ μ such that 2p t t p 2p p t k k k k k k k −1 f (x) = y ∈ η . Therefore x ∈ f (η ). Keeping in view that p = sup Imη ,by t 2p 2p k 2 k k t k −1 −1 Deﬁnition 3.6 we have f (η )(x) = t . That is t ∈ Im f (η ). 2 k k 2 Case II: p < sup Imη . Let p < p in Imη . Then η η . Since Imη is ﬁnite, k 2 k j 2 2p 2p j k we have η η . 2p 2p j k p >p in Imη j k 2 288 Aparna Jain· Naseem Ajmal (2012) Let y ∈ η such that y η . Since f is surjective, choose 2p 2p t k j k p >p in Imη j k 2 −1 −1 x ∈ f (y) ⊆ f (η ). t t 2p k k Suppose, if possible −1 x ∈ f (η ). 2p n n p >p in Imη n k 2 −1 Then, x ∈ f (η ) for some p > p in Imη . By Axiom 4 of an S-morphism, we 2p n k 2 t n −1 −1 −1 have f (η ) ⊆ . f (η ). Thus x ∈ f (η ). This implies t 2p t 2p t 2p n n k n k n y = f (x) ∈ η ⊆ η . t 2p 2 k n p p >p in Imη j k 2 −1 This is a contradiction. And hence x f (η ). 2p t n p >p in Imη n k 2 −1 −1 Therefore x ∈ f (η ) − f (η ). Then by Deﬁnition 3.6 we have 2p 2p t k t n k n p >p in Imη n k 2 −1 t ∈ Im f (η ). k 2 Hence −1 −1 Im f (η ) ⊆ Im f (η ). 1 2 −1 −1 −1 −1 Finally, we show that f (η ) ⊆ f (η ) ∀ t ∈ Im f (η ). Let t ∈ Im f (η ) 1 t 2 t k 1 k 1 k k −1 −1 −1 and let x ∈ f (η ) . Then f (η )(x) ≥ t . We set f (η )(x) = t . Then by 1 t 1 k 1 i Deﬁnition 3.6 −1 −1 x ∈ f (η )andx f (η ). (6) 1p 1p t i t j i j p ≥p in Imη j i 1 t >t in Imμ j i −1 −1 Suppose, if possible f (η )(x) < t . Then t < t , where f (η )(x) = t . Therefore 2 k j k 2 j t < t ≤ t. j k i Since α :Imμ → Im f (μ) is order preserving, we get α(t ) ≤ α(t ). That is p ≤ p in j i j i −1 Imη . Since f (η )(x) = t and t < t , by Deﬁnition 3.6 we get 2 2 j j i −1 x f (η ). (7) t 2p Now, since p ∈ Imη ,η ⊆ η ⊆ η as η η η. i 1 1 2 p 1 2 p p i i i Therefore −1 −1 f (η ) ⊆ f (η ). 1p 2p t i t i i i This by (6) implies −1 x ∈ f (η ), 2p t i −1 −1 which contradicts (7). Hence we have f (η )(x) ≥ t . That is x ∈ f (η ) . 2 k 2 t −1 Therefore∀ t ∈ Im f (η ), we have k 1 −1 −1 f (η ) ⊆ f (η ) . 1 t 2 t k k Fuzzy Inf. Eng. (2012) 3: 273-291 289 Hence −1 −1 f (η ) f (η ). 1 2 In the category S of fuzzy sets, we have the following result, the proof being similar to that of Proposition 4.5 is omitted. Proposition 4.6 Let (X,μ), (Y,η), (Y ,η ), (Y ,η ) ∈ Ob(S), f :(X,μ) → (Y,η) be an 1 1 2 2 S-morphism and (Y ,η ) (Y ,η ) (Y,η). Then we have the following: 1 1 2 2 −1 −1 (i) f (η ) ⊆ f (η ). 1p 2p t k t k k k p ∈Imη p ∈Imη k i k 2 α(t )=p α(t )=p k k k k −1 −1 −1 (ii) f (η ) ⊆ f (η ) ∀ t ∈ Im f (η ). 1 t 2 t k 1 k k Proposition 4.7 Let (X,μ), (Y,η), (Y ,η ) ∈ Ob(S), f :(X,μ) → (Y,η) be an S- −1 morphism and (Y ,η ) (Y,η). Then f ( f (η )) η . Proof It is easy to verify that −1 f ( f (η ) ) ⊆ Y . t t k k −1 t ∈Im f (η ) −1 −1 Now to prove that Im f ( f (η )) ⊆ Imη , let α(t ) ∈ Im f ( f (η )). By Deﬁnition −1 3.5, we have t ∈ Im f (η ). Again by Deﬁnition 3.6, we haveα(t ) ∈ Imη . Hence, k k −1 Im f ( f (η )) ⊆ Imη . −1 −1 Finally, we show that f ( f (η )) ⊆ η ∀ α(t ) ∈ Im f ( f (η )). Let α(t ) k α(t ) −1 −1 α(t ) ∈ Im f ( f (η )) and let y ∈ f ( f (η )) . Then k α(t ) −1 f ( f (η ))(y) ≥ α(t ). −1 Now, if f ( f (η ))(y) = α(t ), then −1 −1 y ∈ f ( f (η ) ) = f ( f (η )) (by Lemma 3.2). t t t i i i t α(t ) i i But −1 f ( f (η )) ⊆ η . i t α(t ) α(t ) i i i Therefore y ∈ η . α(t ) Sinceα(t ) ≥ α(t ), we haveη ⊆ η . That is i k α(t ) α(t ) i k y ∈ η . α(t ) Thus −1 −1 f ( f (η )) ⊆ η ∀ (t ) ∈ Im f ( f (η )). α(t ) k α(t ) k 290 Aparna Jain· Naseem Ajmal (2012) −1 Hence we have f ( f (η )) η . 5. Conclusion This paper attempts to answer questions raised by Goguen by deﬁning and studying a category which has not only its objects as fuzzy but having morphism which are fuzzy as well. Various signiﬁcant properties of this category G have been discussed, but the authors would like to add that it is worthwhile to investigate further properties of this category. Some of these properties would be to verify if products exists in G or if G is algebraic. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Sep 1, 2012
Keywords: Category; Fuzzy set; Subobject; Fuzzy subset; Monomorphism; Image; preimage