sosagt.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 5, No. 2, 2004 pp. 137-154 A fuzzification of the category of M-valued L-topological spaces Tomasz Kubiak and Alexander P. Šostak Abstract. A fuzzy category is a certain superstructure over an ordi- nary category in which ”potential” objects and ”potential” morphisms could be such to a certain degree. The aim of this paper is to introduce a fuzzy category FT OP(L, M) extending the category T OP(L, M) of M-valued L- topological spaces which in its turn is an extension of the category T OP(L) of L-fuzzy topological spaces in Kubiak-Šostak’s sense . Basic properties of the fuzzy category FT OP(L, M) and its objects are studied. 2000 AMS Classification: Primary: 54A40; Secondary: 03E72, 18A05. Keywords: M-valued L-topology, (L, M)-fuzzy topology, L-fuzzy category, GL-monoid, Power-set operators, (L, M)-interior operator, (L, M)-neighborhood system. Introducion The concept of an L-fuzzy topological space, that is of a pair (X, T ) where X is a set and T : LX → L is a mapping subjected to certain axioms was introduced (independently) by T.Kubiak [5] and A. Šostak [9] (Actually a pro- totype of this definition can be traced already in U.Höhle’s work [1].) In some cases it seems reasonable to allow different lattices for domain and codomains of T , resp. L and M, thus coming to the concept of an M-valued L-fuzzy topology on X (or an (L, M)-fuzzy topology on X for short), as a mapping T : LX → M subjected to certain axioms. A detailed study of (L, M)-fuzzy topological spaces will be presented in [6], [7]. In a series of papers the second named author considered the concept of a fuzzy category and the problem of fuzzifications of usual categories (see e.g. [11], [12], [13] etc.) Actually, a fuzzy category is an ordinary category modified in such a way, that ”potential” objects and ”potential” morphisms are such only to a certain degree, and this degree can be any element of the corresponding 138 T. Kubiak and A.Šostak lattice. The concept of a fuzzy category lead us to the idea of ”fuzzification” of some known categories - that is to construct fuzzy categories on the basis of some standard categories. In particular, in [13] we studied fuzzification of some categories related to topology and algebra. It is the aim of this paper to ”fuzzify” the category T OP (L, M) of (L, M)- fuzzy topological spaces. As a tool for this fuzzification we use the structure of a GL-monoid on the codomain lattice (that is lattice M in our cotext), and in particular, the corresponding residuation in it. The structure of the paper is as follows. After introducing the fuzzy cat- egory FT OP (L, M) and other basic definitions in Section 2 we discuss the lattice properties of the family of (L, M)-fuzzy topologies on a set X for a fixed level α (Section 3). Further, in Section 4, we proceed to the study of power- set operators in the context of (L, M)-fuzzy topologies, which, appear to be a convenient and powerfull tool for the investigation of such structures. In Section 5 we consider basic constructions in the fuzzy category FT OP (L, M) of (L, M)-fuzzy topological spaces — namely, products, subspaces, direct sums and quotients. Sections 6 and 7 deal with the inner structure of (L, M)-fuzzy topologies. Namely, in Section 6 we discuss relations between a structure which satisfies the axioms of an (L, M)-fuzzy topology at a level α and the corre- sponding fuzzy interior operator. Further, in Section 7 the relations between this fuzzy interior operator and the corresponding neighbourhood system are discussed. 1. Preliminaries Let L = (L1, ≤L, ∧L, ∨L, ∗L) and M = (M, ≤M, ∧M , ∨M , ∗M ) be GL- monoids (cf e.g. [2], [3]). Let ⊤L, ⊤M and ⊥L, ⊥M denote the top and the bottom elements of L and M respectively. In what follows we shall usually omit the subscripts L and M since from the context it will be clear in what lattice the operation is applied. It is well known that every GL-monoid L is residuated, i.e. there exists a further binary operation — implication ” ֌ ” connected with ∗ by the Galois coonection: α ∗ β ≤ γ ⇐⇒ α ≤ β ֌ γ ∀α, β, γ ∈ L. Let X be a set and LX be the family of all L-subsets of X, i.e. mappings A : X → L. Then all operations on L in an obvious way can be pointwise extended to LX thus generating the structure of a GL-monoid on LX . In particular, implication A ֌ B ∈ LX for L-sets A, B ∈ LX is defined by (A ֌ B)(x) := A(x) ֌ B(x); the top 1X and the bottom elements 0X in L X are defined respectively as 1X(x) = ⊤L ∀x ∈ X and 0X(x) = ⊥L ∀x ∈ X. To recall the concept of an L-valued or L−fuzzy category [11, 12], consider an ordinary (classical) category C and let ω : Ob(C) → L and µ : Mor(C) → L be L−fuzzy subclasses of the classes of its objects and morphisms respectively. Now, an L−fuzzy category can be defined as a triple (C, ω, µ) satisfying the following axioms ([12], cf also [11] in case ∗ = ∧): A fuzzification of the category of M-valued L-topological spaces 139 10 µ(f) ≤ ω(X) ∧ ω(Y ) ∀ X, Y ∈ Ob(C) and ∀f ∈ Mor(X, Y ); 20 µ(g ◦ f) ≥ µ(f) ∗ µ(g) whenever the composition g ◦ f is defined; 30 µ(eX ) = ω(X) where eX : X → X is the identity morphism. 2. Basic definitions Definition 2.1. [M-Fuzzy Category FT OP (L, M).] Let C(L,M) be an (ordinary) category whose objects are pairs (X, T ) where X is a set and T : LX → M is a mapping, and whose morphisms f : (X, TX ) → (Y, TY ) are arbitrary mappings f : X → Y . Given a set X and a mapping T : LX → M we define three fuzzy predicates: ω1(T ) = T (1X ) ( or, equivalently ω1(T ) = ⊤ ֌ T (1X )); ω2(T ) = ∧ U⊂LX,|U|<ℵ0 ( ∧ U∈U T (U) ֌ T ( ∧ U∈U U ) ) ; ω3(T ) = ∧ U⊂LX ( ∧ U∈U T (U) ֌ T ( ∨ U∈U U ) ) . Let ω(T ) = ω1(T ) ∧ ω2(T ) ∧ ω3(T ). Given (X, TX ), (Y, TY ) and a mapping f : X → Y we set ν(f) = ∧ V∈LY ( TY (V ) ֌ TX (f −1(V )) ) , and µ(f) = ν(f) ∧ ωX (TX ) ∧ ωY (TY ). A mapping f will be called continuous if ν(f) = ⊤. Actualy this means that TY (V ) ≤ TX (f −1(V )) for all V ∈ LX. It is easy to note that µ(eX ) = ω(X). Further, if f : (X, TX ) → (Y, TY ) and g : (Y, TY ) → (Z, TZ ) are mappings, then ν(g ◦ f) = ∧ W∈LZ ( TZ (W ) ֌ TX (f −1(g−1(W )) ) ) ≥ ≥ ∧ W∈LZ ( ( TZ(W ) ֌ TY (g −1(W )) ) ∗ ( TY (g −1(W ) ֌ TX (f −1(g−1(W )) ) ) ≥ ∧ W∈LZ (TZ (W ) ֌ TY (g −1(W )) ∗ ∧ V∈LY ( TY (V ) ֌ TX (f −1(V )) ) = = ν(g) ∗ ν(f), and hence also µ(g ∗ f) ≥ µ(g) ◦ µ(f). Thus we arrive at a (M-)fuzzy category FT OP (L, M) = (C(L,M), ω, µ). We interpret ω(T ) as the degree to which a mapping T is an (L, M)-fuzzy topology on X. In case ω(T ) ≥ α we say that T is an (L, M)-fuzzy α-topology on X. An (L, M)-fuzzy ⊤-topology is just an (L, M)-fuzzy topology [6], [7] (and an L-fuzzy topology in case M = L, see e.g. [5], [9], [10]). On the other hand any mapping T : LX → M is an (L, M)-fuzzy ⊥-topology on a set X. 140 T. Kubiak and A.Šostak A pair (X, T ) where T is an (L, M)-fuzzy ⊥-topology will be referred to as an (L, M)-fuzzy ⊥-topological space. Remark 2.2. Applying ω3 to U = ∅ we get ω3(T ) ≤ ⊤ ֌ T (0X ) = T (0X ). Remark 2.3. • ω1(T ) = ⊤ iff T (1X) = ⊤; • ω2(T ) = ⊤ iff ∀ U1, U2 ∈ L X it holds T (U1 ∧ U2) ≥ T (U1) ∧ T (U2); • ω3(T ) = ⊤ iff ∀ U ⊂ L X it holds T ( ∨ U∈U U) ≥ ∧ U∈U T (U). Thus the fuzzy predicates ω1, ω2, ω3 are fuzzifications of the corresponding ax- ioms of an (L, M)-fuzzy topology, cf [6], [7]. Fuzzy predicate ν can be viewed as a version of fuzzification of the axiom of continuity while µ ”touch it up” in order to take into account the ”defectiveness of topologiness” of T . Remark 2.4. [ The case of an idempotent α. ] Let α ∈ L be idempotent, i.e. α ∗ α = α, and let FαT OP (L, M) denote the subcategory of FT OP (L, M) whose objects (X, T ) and morphisms f satisfy conditions ω(T ) ≥ α and µ(f) ≥ α. Then FαT OP (L) is obviously a usual (crisp) category. In particular, F⊤T OP (L, M) = T OP (L, M). Definition 2.5. Given an object (X, T ) of FT OP (L, M), we define a mapping ΣT := Σ : L X → M by setting Σ(A) = T (A ֌ 0X ) for every A ∈ L X. The mapping Σ thus defined is called the degree of closedness in the space (X, T ). Proposition 2.6. [ Basic properties of Σ ] (1) σ1(Σ) := Σ(0X ) = T (1X ) and hence σ1(Σ) = ω1(T ); (2) σ2(Σ) := ∧ A⊂LX ,|A|<ℵ0 ( ∧ A∈A Σ(A) ֌ Σ ( ∨ A∈A A ) ) ≥ ω2(T ) (3) σ3(Σ) := ∧ A⊂LX ( ∧ A∈A Σ(A) ֌ Σ ( ∧ A∈A A ) ) ≥ ω3(T ). Proof. σ1(Σ) = Σ(0X ) = T (0X ֌ 0X ) = T (1X ) = ω1(T ); σ2(Σ) : = ∧ A⊂LX |A|<ℵ0 ( ∧ A∈A Σ(A) ֌ Σ( ∨ A∈A A) ) = = ∧ A⊂LX |A|<ℵ0 ( ∧ A∈A T (A ֌ 0X ) ֌ T ( ( ∨ A∈A A) ֌ 0X ) ) ) = = ∧ A⊂LX |A|<ℵ0 ( ∧ A∈A T (UA) ֌ T ( ∧ A∈A UA) ) ≥ ≥ ∧ U⊂LX |U|<ℵ0 ( ∧ U∈U T (U) ֌ T ( ∧ U∈U U) ) = ω2(T ), A fuzzification of the category of M-valued L-topological spaces 141 where UA := A ֌ 0X. In a similar way, σ3(Σ) := ∧ A⊂LX ( ∧ A∈A Σ(A) ֌ Σ ( ∧ A∈A A ) ) = = ∧ A⊂LX ( ∧ A∈A T (A ֌ 0X ) ֌ T ( ∧ A∈A (A ֌ 0X) ) = = ∧ A⊂LX ( ∧ A∈A T (UA) ֌ T ( ∨ A∈A UA) ) ≥ ≥ ∧ U⊂LX ( ∧ U∈U T (U) ֌ T ( ∨ U∈U U) ) . � Reasoning in a similar way it is easy to establish the following Proposition 2.7. Given a mapping Σ : LX → M let M-valued predicates σ1(Σ), σ2(Σ) and σ3(Σ) be defined as in Proposition 2.6, and let T := TΣ be defined by T (A) = Σ(A ֌ 0X ). Then ω1(T ) = σ1(Σ), ω2(T ) ≥ σ2(Σ), ω3(T ) ≥ σ3(Σ). In case when L is an MV -algebra the L-powerset LX also is an MV -algebra, and hence (A ֌ 0X) ֌ 0X = A for every A ∈ L X . Therefore it follows: Proposition 2.8. If L is an MV -algebra, then TΣT = T and ΣTΣ = Σ. In particular the structures T and Σ mutually define one another. Besides, σ1 ( ΣT ) = ω1(T ), σ2 ( ΣT ) = ω2(T ), σ3 ( ΣT ) = ω3(T ). 3. Lattice properties of (L, M)-fuzzy α-topologies Let α ∈ M be fixed and let Tα(X) := Tα(L, M, X) be the family of all (L, M)-fuzzy α-topologies on a set X. Theorem 3.1. Tα(X) is a complete lattice. Proof. First, notice that Tdis : L X → M defined by Tdis(U) = ⊤ for all U ∈ L X (the so called discrete (L, M)-fuzzy topology) is the top element of Tα(X) and Tind : L X → M defined by Tind(0X ) = Tind(1X ) = α and Tind(U) = ⊥ for U ∈ LX \ {0X, 1X} (the so called indiscrete (L, M)-fuzzy topology) is the bottom element of Tα(X). Further, let T 0 α(X) ⊂ Tα(X) and let T0 : L X → M be defined by the equality T0(U) = ∧ T∈T0α(X) T (U) ∀U ∈ LX. Then ω1(T0) = T0(1X ) = ∧ T∈T0α(X) T (1X ) = ∧ T∈T0α(X) T (1X ) ≥ α; 142 T. Kubiak and A.Šostak ω2(T0) = ∧ U⊂LX |U|<ℵ0 ( ∧ U∈U T0(U) ֌ T0 ( ∧ U∈U U ) ) = = ∧ U⊂LX |U|<ℵ0 ( ∧ U∈U ( ∧ T∈T0α(X) T ) (U) ֌ ∧ T∈T0α(X) ( T ( ∧ U∈U U )) ) ≥ ≥ ∧ T∈T0 α (X) ( ∧ U⊂LX |U|<ℵ0 ( ∧ U∈U T (U) ֌ T ( ∧ U∈U U) ) ) = ∧ T∈T0 α (X) ω2(T ) ≥ α. Reasoning in a similar way we get: ω3(T0) = ∧ U⊂LX ( ∧ U∈U T 0(U) ֌ T0( ∨ U∈U U) ) ≥ ≥ ∧ T∈T0 α (X) ( ∧ U∈LX ( ( ∧ U∈U T (U) ) ֌ T ( ∨ U∈U U) ) ) = ∧ T∈T0 α (X) ω3(T ) ≥ α. Thus T0 ∈ T 0 α(X) and hence T0 is indeed the minimal element of T 0 α(X) in Tα(X). � The previous theorem allows also to write an explicite formula for the supre- mum of a subset T0α(X) ⊂ Tα(X). Namely sup T0α(X) = ∧ {T ∈ Tα(X) | T ≥ Tλ ∀Tλ ∈ T 0 α(X)}. Remark 3.2. Let S : LX → M be a mapping and let the mapping TS : L X → M be defined by TS = ∧ {T : T ∈ Tα(X) and T ≥ S}, where as before Tα(X) := Tα(L, M, X). From Theorem 3.1 it follows that TS is an (L, M)-fuzzy α-topology, besides it is the smallest one (≤) of all (L, M)- fuzzy α-topologies which are greater or equal than S. In this case S is called a subbase of the (L, M)-fuzzy α-topology TS. Proposition 3.3. [ Level decomposition of (L, M)-fuzzy -topologies ] Let T : LX → M be an (L, M)-fuzzy α-topology and assume that γ ∈ M is such that γ ∗ α = γ. Further, let Tγ = {U | T (U) ≥ γ}. Then Tγ is a (Chang-Goguen) L-topology on X. In particular, if α is idempotent, then Tα is a (Chang-Goguen) L-topology on X. Proof. Since ω1(T ) = ⊤ it follows that T (1X ) ≥ α ≥ γ, and 1X ∈ Tγ. Let U1, . . . , Un ∈ Tγ. Then, since ω2(T ) ≥ α, it holds γ ֌ T (U1 ∧ . . . ∧ Un) ≥ T (U1) ∧ . . . ∧ T (Un) ֌ T (U1 ∧ . . . ∧ Un) ≥ α and hence T (U1 ∧ . . . ∧ Un) ≥ α ∗ γ = γ. In a similar way, taking into account that ω3(T ) ≥ α, it is easy to verify that if Ui ∈ Tγ for all i ∈ I, then T ( ∨ i∈I Ui ) ≥ α ∗ γ = γ. � A fuzzification of the category of M-valued L-topological spaces 143 Theorem 3.4. Let S : LX → M be an (L, M)-fuzzy β-topology where α∗β = α Then the mapping T : LX → M defined by T (U) = α ֌ S(U) for every U ∈ LX is an (L, M)-fuzzy topology on X. Proof. Notice first that in this case α = α ∗ β ≤ β, and hence S(1X ) ≥ α. Therefore ω1(T ) = T (1X ) = α ֌ S(1X ) = α ֌ α = ⊤. To verify axioms 2 and 3 for T notice first that for every γ ∈ M it holds α ֌ γ ∗ β = α ֌ γ. Indeed, α ֌ γ = ∨ {λ | λ ∗ α ≤ γ} ≤ ≤ ∨ {λ | λ ∗ α ∗ β ≤ γ ∗ β} = ∨ {λ | λ ∗ α ≤ γ ∗ β} = α ֌ γ ∗ β. The converse inequality is obvious. We proceed as follows. Since ω2(S) ≥ β, we get S( n ∧ i=1 Ui) ≥ n ∧ i=1 S(Ui) ∗ β and hence α ֌ S ( n ∧ i=1 Ui ) ≥ α ֌ n ∧ i=1 S(Ui) ∗ β = α ֌ n ∧ i=1 S(Ui) = n ∧ i=1 ( α ֌ S(Ui) ) ; thus T ( ∧n i=1 Ui) ≥ ∧n i=1 T (Ui). From ω3(S) ≥ β, reasoning in a similar way as above, we conclude that S ( ∨ i∈I Ui ) ≥ ∧ i∈I S(Ui) ∗ β, and hence T ( ∨ i∈I Ui) ≥ ∧ i∈I T (Ui) for any family {Ui | i ∈ I } ⊂ L X. � Corollary 3.5. If S : LX → M is an (L, M)-fuzzy α-topology and α is idem- potent, then the mapping T : LX → M defined by T (U) := α ֌ S(U) for every U ∈ LX is an (L, M)-fuzzy topology. If S : LX → M is an (L, M)-fuzzy topology then for every α the mapping T (U) = α ֌ S(U) is an (L, M)-fuzzy topology. Theorem 3.6. Let f : (X, TX ) → (Y, TY ) be a mapping, ω(TX ) ≥ β, ω(TY ) ≥ α where β ∗ α = α, and let SY : L Y → M be a subbase of TY . Then the following conditions are equivalent: 10 TY (V ) ֌ TX (f −1(V )) ≥ α ∀V ∈ LY ; 20 SY (V ) ֌ TX (f −1(V )) ≥ α ∀V ∈ LY . In particular, these conditions are equivalent in case when α ≤ β and α is idempotent. 144 T. Kubiak and A.Šostak Proof. Since TY (V ) ≥ SY (V ), it holds TY (V ) ֌ TX (f −1(V )) ≤ SY (V ) ֌ TX (f −1(V )) and hence 10 =⇒ 20. Conversely, if SY (V ) ֌ TX (f −1(V )) ≥ α for all V ∈ LY , then SY (V ) ≤ α ֌ TX (f −1(V )) ∀V ∈ LX. Let now T ′(V ) := TX (f −1(V )). It is easy to verify that T ′ is an (L, M)-fuzzy β- topology since TX is an (L, M)-fuzzy β-topology. Further, let T ′′ : LY → M be defined by T ′′(V ) := α ֌ T ′(V ). Then by Theorem 3.4 T ′′ is an (L, M)-fuzzy topology on Y . Moreover, SY (V ) ≤ T ′′(V ). Thus, since TY is an (L, M)-fuzzy α-topology generated by subbase SY , it follows that SY (V ) ≤ TY (V ) ≤ T ′′(V ), and hence TY (V ) ≤ α ֌ TX (f −1(V )) =⇒ α ≤ TY (V ) ֌ TX (f −1(V )) ∀V ∈ LY . � Question 3.7. Do the statements of Corollary 3.5 and Theorem 3.6 hold also in case α = β but without assumption of idempotency of α ? 4. Power-set operators and (L, M)-fuzzy α-topologies Let X, Y be sets, and let F : LY → LX be a mapping preserving arbitrary joins and meets. In particular, F (1Y ) = 1X and F (0Y ) = 0X. Definition 4.1. (cf e.g. [8]) The powerset operator F→ : M(L Y ) → M(L X) of a mapping F : LY → LX is defined by the equality F→(TY )(U) = ∨ {TY (V ) : F (V ) = U}, ∀U ∈ L X for every TY : L Y → M, Definition 4.2. (cf e.g [8]) The powerset operator F← : M(L X) → M(L Y ) of a mapping F : LY → LX is defined by the equality F ←(TX )(V ) = TX ( F (V ) ) ∀V ∈ LY for every TX : L X → M, The following two theorems show that the powerset operators F→ and F← do not diminish the topologiness degree of the mappings TY and TX respec- tively. Theorem 4.3. If M is completely distributive, then ω(F→(TY )) ≥ ω(TY ) := α. ( Actually, ω1(F →(TY )) ≥ ω1(TY ), ω2(F →(TY )) ≥ ω2(TY ) and ω3(F →(TY )) ≥ ω3(TY ). ) A fuzzification of the category of M-valued L-topological spaces 145 Proof. Since F→(TY )(1X ) = ∨ {TY (V ) | F (V ) = 1X} ≥ TY (1Y ) ≥ α, it follows that ω1(F →(TY )) ≥ α. To verify that ω2(F →(TY )) ≥ α fix some U1, . . . Un ∈ L X and let U0 := ∧n i=1 Ui. We have to show that ( n ∧ i=1 F→(TY )(Ui) ) ֌ F→(TY )(U0) ≥ α. If for some i ∈ {1, . . . , n} there does not exist Vi ∈ L Y such that F (Vi) = Ui, then from the definition of F→(TY ) it is clear that F →(TY )(Ui) = ⊥ and hence the inequality is obvious. Assume therefore that for each i = 1, . . . , n some Vi ∈ L Y is fixed such that Ui = F (Vi). Then, since ω2(TY ) ≥ α, and since F ( n ∧ i=1 Vi) = n ∧ i=1 F (Vi) = ∧ Ui = U0 it follows that n ∧ i=1 TY (Vi) ֌ TY ( n ∧ i=1 Vi) ≥ n ∧ i=1 TY (Vi) ֌ ∨ V0∈L Y F (V0)=U0 TY (V0) ≥ α. This holds for any choice of Vi ∈ L Y , i ∈ {1, . . . , n}, satisfying F→(Vi) = Ui, and therefore taking into account that L is infinitely distributive, we conclude that n ∧ i=1 ( F→(TY )(Ui) ) ֌ F→(TY )(U0) = n ∧ i=1 ( ∨ F (Vi)=Ui Vi∈LY TY (Vi) ) ֌ ∨ F (Vi)=Ui V0∈LY TY (V0) ) = ( ∨ F (Vi)=Ui n ∧ i=1 TY (Vi) ) ֌ ∨ F (V0)=U0 V0∈LY TY (V0) ≥ α. To verify the third inequality, ω3(F →(TY )) ≥ α, fix a family U = {Ui | i ∈ I}, and let U0 := ∨ i∈I Ui. We have to show that ∧ i∈I F→(TY )(Ui) ֌ F →(TY )( ∨ i∈I Ui) ≥ α. Let for each i ∈ I an L-set Vi ∈ L Y be fixed such that F (Vi) = Ui. (As in the previous situation it is sufficient to assume that such choice of Vi ∈ L Y for all i ∈ I is possible.) Then ∧ i∈I TY (Vi) ֌ TY ( ∨ i∈I Vi) ≥ α, that is ∧ i∈I TY (Vi) ≥ TY ( ∨ i∈I Vi) ∗ α. 146 T. Kubiak and A.Šostak Applying complete distributivity we get the following chain of (in)equalities: α ∗ F→(TY )( ∨ i∈I Ui) = α ∗ ∨ {TY ( ∨ i∈I Vi) : F (∨iVi) = ∨iUi} = = ∨ ( α∗{TY ( ∨ i∈I Vi) : F (∨iVi) = ∨iUi} ) ≥ ∨ { ∧ i∈I TY (Vi) : F (∨iVi) = ∨iUi} ≥ ≥ ∨ { ∧ i∈I TY (Vi) : F (Vi) = Ui} = ∨ { ∧ i∈I TY (Vi) : Vi ∈ Vi := {V | Ui = F (V )}} = = ∨ ϕ∈ ∏ i Vi ( ∧ i∈I TY (ϕ(i))) = ∧ i∈I ∨ Vi∈Vi TY (Vi) = ∧ i∈I ∨ F (Vi)=Ui TY (Vi) = ∧ i∈I F→(TY )(Ui). and hence we obtain the required inequality: F→(TY )( ∨ i∈I Ui) ֌ ∧ i∈I F→(TY )(Ui) ≥ α. � Theorem 4.4. ω(F←(TX )) ≥ ω(TX ) =: α. ( Actually, ω1(F ←(TX )) = ω1(TX ), ω2(F ←(TX )) ≥ ω2(TX ) and ω3(F ←(TX )) ≥ ω3(TX ). ) Proof. ω1(F ←(TX )) = F ←(TX )(1Y ) = TX (F (1Y )) = TX (1X ) = ω1(TX ) ≥ α. To verify condition ω2(TY ) ≥ α, where TY := F ←(TX ), fix {V1, . . . , Vn} ⊂ L Y , then n ∧ i=1 TY (Vi) ֌ TY ( n ∧ i=1 Vi ) = n ∧ i=1 F←(TX )(Vi) ֌ F ←(TX ) ( n ∧ i=1 Vi ) = n ∧ i=1 TX ( F (Vi) ) ֌ TX ( n ∧ i=1 F (Vi) ) ≥ α. Finally, to verify the condition ω3(TY ) ≥ α fix a family V = {Vi | i ∈ I} ⊂ L Y . Then ∧ i∈I TY (Vi) ֌ TY ( ∨ i∈I (Vi) ) = ∧ i∈I F←(TX )(Vi) ֌ F ←(TX )( ∨ i∈I Vi) = = ∧ i∈I TX (F (Vi)) ֌ TX ( ∨ i∈I F (Vi)) ≥ ω3(TX ) ≥ α. � Power-set operators F← and F→ can be applied, in particular, for descrip- tion of final and initial (L, M)-fuzzy α-topologies. Here are some details: Let f : X → Y be a mapping, then by setting f←(V ) := f−1(V ) one defines a mapping f← : LY → LX, which obviously, preserves joins and meets, and so one can apply to it theorems 4.3 and 4.4. Namely, one can get the following corollaries from the statements of these theorems and from the definition of power-set operators. A fuzzification of the category of M-valued L-topological spaces 147 Corollary 4.5. Let TY : L Y → M be a mapping where M is completely distributive, and let ω(TY ) ≥ α. Then given a mapping f : X → Y , it holds ω ( (f←) → (TY ) ) ≥ α. Besides, (f←) → (TY ) is the weakest (L, M)-fuzzy α-topology (actually, even the weakest (L, M)-fuzzy ⊥-topology!) on X for which the mapping f : (X, TX ) → (Y, TY ) is continuous (i.e. ν(f) = ⊤). Corollary 4.6. Let TX : L X → M be a mapping and ω(TX ) ≥ α. Then given a mapping f : X → Y it holds ω ( (f←) ← (TX ) ) ≥ α. Besides, (f←) ← (TX ) is the strongest (L, M)-fuzzy α-topology (actually, even the strongest (L, M)- fuzzy ⊥-topology!) on Y for which f : (X, TX ) → (Y, TY ) is continuous (i.e. ν(f) = ⊤). 5. Products, subspaces, direct sums and quotients In this section we shall discuss how basic operations for (L, M)-fuzzy α- topological spaces can be defined. 5.1. Products. Let X = {(Xi, Ti) : i ∈ I} be a family of (L, M)-fuzzy α- topological spaces, where M is completely distributive, let X = ∏ i∈I Xi be the product of the corresponding sets, and let pi : X → Xi be the projections. Further, let T̂i := (p ← i ) →(Ti) : L X → M. Then, by Corollary 4.5, ω(T̂i) ≥ α. Let S := ∨ i∈I T̂i and let TX : L X → M be the (L, M)-fuzzy α-topology generated by the subbase S : LX → M. Then, obviously, TX is the weakest (L, M)-fuzzy α-topology for which all projections are continuous (i.e. ν(pi) = ⊤). Moreover, the pair (X, TX ) is the product of the family X in the fuzzy category FT OP (L, M) in the following sense: Given an (L, M)-fuzzy β-topological space (Z, TZ ) where β∗α = α, and a family of mappings fi : (Z, TZ ) → (Xi, Ti), i ∈ I, there exists a unique mapping h : (Z, TZ) → (X, TX ) such that pi ◦ h = fi for all i ∈ I and ν(h) ≥ α ⇐⇒ ∧ i∈I ν(fi) ≥ α. Indeed, let h := △i∈Ifi : Z → X be the diagonal product of mappings fi, i ∈ I. If ν(h) ≥ α, then for every i ∈ I ν(fi) = ν(pi ◦ h) ≥ ν(pi) ∗ ν(h) ≥ ⊤ ∗ ν(h) ≥ α. Conversely, let ν(fi) ≥ α for all i ∈ I. We have to verify that in this case TX (W ) ֌ TZ (h −1(W )) ≥ α ∀W ∈ LX. According to Theorem 3.6 it is sufficient to verify that S(W ) ֌ TZ (h −1(W )) ≥ α ∀W ∈ LX. However, from the definition of S it is clear that S(W ) = Ti(Vi) if W := Ṽi where Ṽi = p −1 i (Vi) for some Vi ∈ L Xi and S(W ) = ⊥ otherwise. Therefore 148 T. Kubiak and A.Šostak it is sufficient to verify the above inequality for L-sets of the form Ṽi. How- ever, in this case h−1(Ṽi) = h −1(p−1i (Vi)) = f −1 i (Vi), and hence the requested inequality can be rewritten as Ti(Vi) ֌ TZ(f −1 i (Vi)) ≥ α which holds according to our assumptions. 5.2. Subspaces. Let (X, T ) be an (L, M)-fuzzy α-topological space, let X0 ⊂ X and let e : X0 → X be the embedding mapping. Further, let T0 := (e←)→(T ). Then according to Corollary 4.5 ω(T0) ≥ ω(T ) and hence (X0, T0) is an (L, M)-fuzzy α-topological space. From the construction it is clear that ν(e) = ⊤. Moreover, it is easy to note that (X0, T0) is a subobject of (X, T ) in the following sense: For every (L, M)-fuzzy ⊥-topological space (Z, TZ ) and for every mapping f : (Z, TZ) → (X0, T0) it holds ν(f) = ν(e ◦ f). Indeed, let V0 = e −1(V ) for some V ∈ LX. Then T0(V0) ֌ TZ (f −1(V0)) ≥ T (V ) ֌ TZ (f −1(e−1(V ))) = = T (V ) ֌ TZ((e ◦ f) −1(V )) ≥ ν(e ◦ f), and hence ν(f) ≥ ν(e ◦ f). The converse inequality is obvious. 5.3. Coproducts (Direct sums). Let X = {(Xi, Ti) : i ∈ I} be a family of (L, M)-fuzzy α-topological spaces, let X = ⊕i∈IXi be the disjoint union of the corresponding sets, and let ei : Xi → X be the inclusion mapping. Further, let Si := (e ← i ) ←(Ti). Then by Corollary 4.6 ω(Si) ≥ α, and hence, according to Theorem 3.1 T := ∧ i∈I Si is an (L, M)-fuzzy α-topology. Besides, it is clear that T is the strongest (L, M)-fuzzy α-topology for which all inclusions ei are continuous, i.e. ν(ei) = ⊤. Moreover (X, T ) is the coproduct of the family X in the fuzzy category FT OP (L, M) in the following sense: Let (Z, TZ ) be an (L, M)-fuzzy ⊥-topological space and let fi : (Xi, Ti) → (Z, TZ ), i ∈ I, be a family of mappings. Further, let the mapping f : (X, T ) → (Z, TZ ) be defined by f(x) = fi(x) iff x ∈ Xi. Then ν(f) = ∧ i∈I ν(fi). Indeed, since fi = f ◦ ei and ν(ei) = ⊤ for all i ∈ I, the inequality ν(f) ≥ ∧ i∈I ν(fi) is obvious. Conversely, assume that ∧ i∈I ν(fi) ≥ α. Then α ≤ TZ(V ) ֌ Ti(f −1(V )) = TZ(V ) ֌ Si(ei(f −1 i (V )) = TZ (V ) ֌ Si(f −1(V )). A fuzzification of the category of M-valued L-topological spaces 149 Now, taking infimum over all i ∈ I, we obtain: TZ(V ) ֌ T (f −1(V )) ≥ α. 5.4. Quotients. Let (X, TX ) be an (L, M)-fuzzy α-topological space and let f : X → Y be a surjective mapping. Further, let TY = (f ←)←(TX ). Then, according to Corollary 4.6 ω(TY ) ≥ α and hence (Y, TY ) is an (L, M)-fuzzy α- topological space. It is clear that TY is the strongest (L, M)-fuzzy α-topology for which the mapping f is continuous, i.e. ν(f) = ⊤. The pair (Y, TY ) can be viewed as the quotient of (X, TX ) under mapping f in the fuzzy category FT OP (L, M) in the following sense: Let (Z, TZ ) be an (L, M)-fuzzy α-topological space and let g : (Y, TY ) → (Z, TZ ) be a mapping. Then ν(g ◦ f) = ν(g). Indeed, the inequality ν(g ◦ f) ≤ ν(g) holds always. To establish the converse inequality let h = g ◦ f and let W ∈ LZ. Then by surjectivity of the mapping f there exists U ∈ LX such that g−1(W ) = f(U) and, in particular, U = f−1(g−1(W )). Hence, by definition of TY we have TY (g −1(W )) = TX (f −1(g−1(W ))) = TX (h −1(W )). It follows from here that TZ (W ) ֌ TY (h −1(W )) = TZ (W ) ֌ T (g −1(W )) and taking infimum over all W ∈ LZ we obtain: ν(g ◦ f) ≥ ν(g). 6. Interior operator Theorem 6.1. Let T : LX → M be a mapping where M is completely dis- tributive and let ω(T ) ≥ α. We define the mapping Int := IntT : L X × M → LX by setting: Int(A, β) = ∨ {U : U ≤ A, T (U) ≥ β} ∀A ∈ LX, ∀β ∈ M. Then: (1int) Int(1X, β) = 1X ∀β ≤ α; (2int) A ≤ A′, β′ ≤ β =⇒ Int(A, β) ≤ Int(A′, β′); (3int) ∧ i=1,...,n Int(Ai, β) ≤ Int( ∧ i=1,...,n Ai, β ∗ α) ∀β ∈ M; (4int) Int(A, ⊥) = A. (5int) Int(Int(A, β), β ∗ α) ≥ Int(A, β) ∀β ∈ M; (6int) If Int(A, β) = A0 ∀β ∈ M′, then Int(A, ∨M′) = A0. 150 T. Kubiak and A.Šostak Besides, if ω(T ) = ⊤, then Int satisfies the following stronger version of the property (5int): (5int0 ) Int(Int(A, β), β) ≥ Int(A, β) Conversely, if a mapping Int : LX ×M → LX satisfies conditions (1int) - (6int) above for a fixed α ∈ M , then the mapping T := TInt : L X → M defined by the equality T (A) = ∨ {β ∈ M : Int(A, β) = A} is an (L, M)-fuzzy α-topology on X and besides ω3(TInt) = 1. (In the sequel mappings Int : LX × M → LX satisfying the above properties (1int) - (6int) for a fixed α ∈ M will be referred to as an (L, M)-fuzzy α- inte- rior operator.) The (L, M)-fuzzy α-topology and the corresponding (L, M)-fuzzy α-interior op- erator are related in the following way: TIntT ∗ α ≤ T ≤ TIntT and IntTInt ≤ Int and IntTInt (·, β ∗ α) ≥ Int(·, β) ∀β ∈ M. In case ω3(T ) = ⊤, the equalities T = TIntT and Int = IntTInt hold (cf Theorem 8.1.2 in [4]). Proof. (1) Since ω1(T ) ≥ α, it follows that T (1X ) ≥ α ≥ β and hence Int(1X, β) ≥ 1X. (2) Obvious. (3) Applying infinite distributivity of the lattice M and condition ω2(T ) ≥ α we have ∧ i=1...n Int(Ai, β) = ∧ i ( ∨ {Ui | Ui ≤ Ai, T (Ui) ≥ β} ) ≤ ≤ ∨ { ∧ i Ui | Ui ≤ Ai, T (Ui) ≥ β } ≤ ≤ ∨ {V | V ≤ ∧ i Ai, T (V ) ≥ β ∗ α} = = Int( ∧ i Ai, β ∗ α). (4) Obvious. (5) Int(A, β) = ∨ {U ∈ LX | U ≤ A, T (U) ≥ β}; hence by condition ω3(T ) ≥ α we have T (Int(A, β)) ≥ β ∗ α and therefore Int(Int(A, β), β ∗ α) ≥ Int(A, β). (6) Int(A, ∨ M′) = ∨ {U ∈ LX | U ≤ A, T (U) ≥ ∨ M′} = ∨ {U ∈ LX | U ≤ A, T (U) ≥ β ∀β ∈ M′} = A0. Moreover, if ω3(T ) = ⊤, then T (Int(A, β)) ≥ β and hence Int(Int(A, β), β) ≥ Int(A, β). A fuzzification of the category of M-valued L-topological spaces 151 Conversely, (1) if Int(1X, β) = 1X for all β ≤ α, then TInt(1X ) ≥ α, and hence ω1(TInt) ≥ α. (2) Let U1, . . . , Un ∈ L X, and let β0 := TInt(U1) ∧ . . . ∧ TInt(Un). Then for every β ≪ β0 Int(Ui, β) ≥ Ui and hence by property (3 int) Int( n ∧ i=1 Ui, β ∗ α) ≥ n ∧ i=1 Int(Ui, β) ≥ n ∧ i=1 Ui. It follows from here that TInt( n ∧ i=1 Ui) = ∨ {γ | Int( n ∧ i=1 Ui, γ) ≥ n ∧ i=1 Ui} ≥ β ∗ α for every β ≪ β0 and hence, by complete distributivity of M TInt( n ∧ i=1 Ui) ≥ β0 ∗ α. Therefore n ∧ i=1 TInt(Ui) ֌ TInt( n ∧ i=1 Ui) ≥ α, for each finite family {U1, . . . , Un} ⊂ L X and hence ω2(T ) ≥ α. (3) Let U := {Ui | i ∈ I} and let ∧ i∈I TInt(Ui) =: β0. Then for every i ∈ I and for every β ≪ β0 it holds Int(Ui, β) ≥ Ui. Applying (2 int) we conclude from here that ∨ i∈I Ui ≤ ∨ i∈I Int(Ui, β) ≤ Int( ∨ i∈I Ui, β) for every β ≪ β0. and hence TInt( ∨ i∈I Ui) ≥ β. Hence, by complete distribu- tivity of the lattice M we conclude: TInt( ∨ i∈I Ui) ≥ ∧ i∈I TInt(Ui). Thus ω3(TInt) = ⊤ and hence ω(TInt) ≥ α To verify the relations between TIntT and T , take some U ∈ L X and let T (U) =: β. Then IntT (U, β) ≥ U, and hence TIntT (U) ≥ β, thus the inequality T ≤ TIntT is established. Conversely, let TIntT (U) = ∨ {β | IntT (U, β) ≥ U} = β0. Then for each β ≪ β0 IntT (U, β) = ∨ {V |T (V ) ≥ β} = U, and hence, in view of the property ω3(T ) ≥ α, we conclude that T (IntT (U, β)) ≥ β ∗ α. Since this holds for every β ≪ β0 and for every U ∈ L X it follows from here that TIntT ∗ α ≤ T . In particular, if ω3(T ) = ⊤, then TIntT = T . 152 T. Kubiak and A.Šostak Let now A ∈ LX and let Int(A, β) =: W. Then by the property (5int) of the (L, M)-fuzzy α-interior operator and from the definition of the (L, M)- structure TInt : L X → M we have TInt(W ) ≥ β and hence, taking into account monotonicity and property (4int) of the (L, M)-fuzzy α-interior operator, it follows IntTInt (A, β ∗ α) ≥ IntTInt(W, β) ≥ W = Int(A, β), i.e. IntTInt(·, β ∗ α) ≥ Int(·, β). In particular, if ω3(T ) = ⊤, then IntTInt = Int. Conversely, let IntTInt (M, β) =: W, then by the definition of IntTInt , we conclude that ∨ {U|TInt(U) ≥ β, U ≤ A} = W. Taking into account that, as it was already established above, ω3(TInt) = ⊤ it follows that β ≤ TInt(W ) = ∨ {β′ |Int(W, β′) = W}. properties (6int) and (2int) we conclude that Int(A, β) ≥ Int(W, β) ≥ W and hence Int ≥ IntTInt, that is IntTInt(·,β) ≤ Int(·, β) ≤ IntTInt (·, β ∗ α). In particular, if ω3(T ) = ⊤, then Int = IntTInt. � 7. Neighborhood systems Let Int : LX → M be an (L, M)-α-fuzzy interior operator, i.e. Int satisifes properies (1int) — (6int). Theorem 7.1. Let NInt := N : X × L X × L → L be defined by the equality N (x, U, β) = Int(U, β)(x). Then: (1N ) N (x, 1X, β) = ⊤ ∀x ∈ X if β ≤ α; (2N ) U ≤ U′, β′ ≤ β =⇒ N (x, U, β) ≤ N (x, U′, β′); (3N ) ∧n i=1 N (x, Ui, β) ≤ N (x, ∧n i=1 Ui, β ∗ α); (4N ) N (x, U, 0) = Int(U, 0)(x) (= U(x));. (5N ) N (x, U, β) ≤ ∨ {N (x, V, β ∗ α) | V (y) ≤ N (y, U, β) : ∀y ∈ X}; (6N ) if U(x) ≤ N (x, U, β) ∀x ∈ X, ∀β ∈ M′ ⊂ M, then U(x) ≤ N (x, U, ∨M′). Conversely, if N : X × LX × L → L satisfies conditions (1N) — (6N) above, then the mapping IntN := Int : L X × L → LX defined by Int(U, β)(x) = N (x, U, β) satisfies axioms (1int) − (6int), i.e. is an (L, M)-fuzzy α-interior operator. Moreover, IntNInt = Int and NIntN = N . A fuzzification of the category of M-valued L-topological spaces 153 Proof. Let Int : LX × M −→ LX be an (L, M)-fuzzy α-interior operator and let N := NInt be defined as above. (1N): For β ≤ α by (1int) it holds N (x, 1X, β) = Int(1X, β)(x) = 1X (x) = ⊤. (2N): If U ≤ U′ and β′ ≤ β, then by (2int) it holds N (x, U, β) = Int(U, β)(x) ≤ Int(U′, β′)(x) = N (x, U′, β′). (3N): Applying (3int) we get: ∧ i=1,...,n N (x, Ui, β) = ∧ i=1,...,n Int(Ui, β)(x) ≤ ≤ Int( ∧ i=1,...,n Ui, β ∗ α)(x) = N (x, ∧ i=1,...,n Ui, β ∗ α) (4N) obviously follows from (4int). (5N): Applying (5int) and denoting Int(U, β ∗ α) = V we get: N (x, U, β) = Int(U, β)(x) ≤ Int(Int(U, β), β ∗ α)(x) = = Int(V, β ∗ α)(x) ≤ ∨ {Int(W, β ∗ α) | Int(W, β ∗ α) ≤ V } = = ∨ N (x, V, β ∗ α) ≤ ≤ ∨ {N (x, W, β ∗ α)|W (y) ≤ N (y, U, β) ∀y ∈ X}. (6N): Assume that U(x) ≤ N (x, U, β) for every β ∈ M′ and every x ∈ X. Then U(x) = Int(U, β)(x) and hence U = Int(U, β) for every β ∈ M′. Applying property (6int) of the (L, M)-fuzzy α-interior operator we conclude that U = Int(U, ∨ M′) and hence U(x) = N (x, U, ∨ M′). Conversely, let N : X × LX × M → M satisfy the properties (1int) — (6int) and let Int = IntN be defined as above. Then Int is the interior operator. The validity of properties (1int), (2int), (3int), (4int) and (6int) is obvious from the definition of IntN and the corresponding properties of N . To show (5int) notice that Int(U, β)(x) = N (x, U, β) ≤ ∨ {N (x, W, β ∗ α)|W (y) ≤ N (y, U, β)} = = ∨ {Int(W, β ∗ α)(x)|W ∈ LX such that W (y) ≤ Int(U, β)(y)} = Int(Int(U, β), β ∗ α)(x) Finally, the equalities IntNInt = Int and NIntN = N are obvious from the definitions. � References [1] U.Höhle, Uppersemicontinuous fuzzy sets and applications, J. Math. Anal. Appl. 78 (1980), 659-673. [2] U.Höhle, Commutative, residuated l-monoids, In: Non-classical Logics and Their Ap- plications to Fuzzy Subsets: A Handbook of the Mathematical Foundations of Fuzzy Subsets, E.P. Klement and U. Höhle eds., Kluwer Acad. Publ., 1994, 53-106. 154 T. Kubiak and A.Šostak [3] U. Höhle, M-valued sets and sheaves over integral commutative cl-monoids, In: Appli- cations of Category Theory to Fuzzy Sets., S.E. Rodabaugh, E.P. Klement and U. Höhle eds., Kluwer Acad. 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Šostak, On a concept of a fuzzy category, In: 14th Linz Seminar on Fuzzy Set Theory: Non-classical Logics and Applications. Linz, Austria, 1992, pp. 62-66. [12] A. Šostak, Fuzzy categories versus categories of fuzzily structured sets: Elements of the theory of fuzzy categories, In: Mathematik-Arbeitspapiere, Universität Bremen, vol 48 (1997), pp. 407-437. [13] A. Šostak, Fuzzy categories related to algebra and topology, Tatra Mount. Math. Publ. 16:1, (1999), 159-186. Received March 2003 Accepted June 2003 Tomasz Kubiak (tkubiak@amu.edu.pl) Wydzia l Matematyki i Informatyki, Adam Mickiewicz University, PL-60-769, Poznań, Poland Alexander Šostak (sostaks@com.latnet.lv) Departmant of Mathematics, University of Latvia, LV-1586 Riga, Latvia