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Diagonal conditions for lattice-valued uniform convergence spaces
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Fuzzy Sets and Systems 210 (2013) 39 – 53 www.elsevier.com/locate/fss
Diagonal conditions for lattice-valued uniform convergence spaces Gunther Jäger Department of Statistics, Rhodes University, 6140 Grahamstown, South Africa Received 31 December 2011; received in revised form 7 May 2012; accepted 7 May 2012 Available online 17 May 2012
Abstract We define diagonal conditions for lattice-valued uniform convergence spaces and show that these conditions are preserved under initial constructions. Further, the forgetful functor from the category of stratified lattice-valued uniform convergence spaces into the category of stratified lattice-valued limit spaces maps these conditions to corresponding conditions. We especially define uniform regularity and characterize this condition by certain closures of lattice-valued filters. As an application we generalize an extension theorem for uniformly continuous mappings. © 2012 Elsevier B.V. All rights reserved. Keywords: L-fuzzy convergence; L-topology; L-filter; L-uniform convergence space; Diagonal condition; Uniform regularity; Extension theorem
1. Introduction The category of stratified lattice-valued uniform convergence spaces, SL-UCS, was introduced, for L a complete Heyting algebra, in [20]. The lattice context was extended to enriched cl-premonoids in [6,17]. The category SL-UCS is well-fibred and topological over SET and in the case that L is a complete Heyting algebra, it is Cartesian closed. SL-UCS generalizes the classical category of uniform convergence spaces UCS [4,27] in the sense that for L = {0, 1}, S{0, 1}UCS and UCS can be identified. SL-UCS contains the category of lattice-valued uniform spaces [8,9] as a reflective subcategory. Moreover, for a suitable choice of the underlying lattice, the categories of probabilistic uniform limit spaces [24] and of approach uniform spaces [23] are examples of the category of lattice-valued uniform convergence spaces [17]. In this paper, we study certain diagonal conditions in SL-UCS. In particular we define a Fischer-type diagonal condition and characterize it by means of certain neighbourhood operators. Furthermore, a concept of uniform regularity is introduced. This concept generalizes the classical concept studied by Gähler [7] and is “dual” to the Fischer-type diagonal condition. All studied conditions are finally “tied together” in an extension theorem for uniformly continuous mappings. Also this Extension Theorem generalizes a “classical” result of Gähler [7]. The paper is organised as follows. In the first section we fix the lattice background and our notation as well as the lattice-valued filter concepts that we use. In the next section we review lattice-valued uniform convergence spaces and lattice-valued limit spaces. Section 4 is then devoted to diagonal conditions à la Kowalsky and à la Fischer. Section 5 studies the concept of uniform regularity and Section 6 then presents a theorem on extension of uniformly continuous mappings. Finally we draw some conclusions. E-mail address: [email protected] 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.05.001
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
2. Preliminaries We consider in this paper enriched cl-premonoids [11,9] (L , ⱕ , ⊗, ∗). These are complete lattices (L , ⱕ ) with two algebraic operations ⊗, ∗ such that (L , ⱕ , ⊗) is a cl-premonoid, i.e. (E1) 1 ⊗ 1 ⱕ 2 ⊗ 2 whenever 1 ⱕ 2 and 1 ⱕ 2 ; (E2) ⱕ ⊗ and ⱕ ⊗ for all ∈ L; (E3) ⊗ is distributive over non-empty joins, i.e. for J ∅, j , ∈ L ( j ∈ J ): ⎛ ⎝
⎞ j⎠ ⊗ =
j∈J
⎛ ( j ⊗ ) and ⊗ ⎝
j∈J
j∈J
⎞ j⎠ =
( ⊗ j );
j∈J
and (L , ⱕ , ∗) is a GL-monoid, i.e. (E4) (L , ∗) is a commutative semigroup with as unit element; (E5) there is ∈ L such that = ∗ whenever ⱕ (divisibility [10]); (E6) ∗ is distributive over arbitrary joins, i.e. for any set J, j , ∈ L ( j ∈ J ): ⎛ ⎝
j∈J
⎞ j⎠ ∗ =
( j ∗ );
j∈J
and (E7) ∗ is dominated by ⊗, i.e. for all 1 , 2 , 1 , 2 ∈ L: (1 ⊗ 1 ) ∗ (2 ⊗ 2 ) ⱕ (1 ∗ 2 ) ⊗ (1 ∗ 2 ). From (E7) we conclude with 1 = , 2 = , 1 = , 2 = that ∗ ⱕ ⊗ for any , ∈ L. This motivates the term domination. Examples 2.1. • For a GL-monoid (L , ⱕ , ∗), the quadruple (L , ⱕ , ∗, ∗) is an enriched cl-premonoid. As a special case, with ∗ = ∧, a complete Heyting algebra, i.e. a complete lattice (L , ⱕ ) in which the frame law ( i∈J i ) ∧ = i∈J (i ∧ ) holds, is an enriched cl-premonoid, (L , ⱕ , ∧, ∧). • For a complete Heyting algebra (L , ⱕ , ∧) and a GL-monoid (L , ⱕ , ∗), the quadruple (L , ⱕ , ∧, ∗) is an enriched cl-premonoid. • Further examples are given by certain GL-monoids (L , ⱕ ,, ∗ ∗) with a monoidal mean operator ∗ [11]. • A further example is given by triangular norms on [0, 1] (see [26,21]). A t-norm is a binary operation ∗ : [0, 1] × [0, 1] −→ [0, 1] which is commutative, associative, non-decreasing in both arguments and has 1 as the unit. Important examples are the minimum t-norm, ∗ = ∧ , the product t-norm, ∗ = · and the Lukasiewicz t-norm, ∗ = ( + − 1) ∨ 0. It is clear, that for a t-norm ∗ which is left-continuous (as a function from [0, 1]2 to [0, 1]), ([0, 1], ⱕ , ∗) is a GL-monoid. All the above mentioned t-norms have monoidal mean operators such that ([0, 1], ⱕ ,, ∗ ∗) as well as ([0, 1], ⱕ , ∧, ∗) are enriched cl-premonoids. All of the enriched cl-premonoids (L , ⱕ , ⊗, ∗) mentioned above are pseudo-bisymmetric [11], i.e. for all 1 , 2 , 1 , 2 ∈ L we have (1 ∗ 1 ) ⊗ (2 ∗ 2 ) ⱕ ((1 ⊗ 2 ) ∗ (1 ⊗ 2 )) ∨ ((1 ⊗ ⊥) ∗ (1 ⊗ )) ∨ ((⊥ ⊗ 2 ) ∗ ( ⊗ 2 )). Let (L , ⱕ , ⊗, ∗) be an enriched cl-premonoid and X be a set. We denote the L-sets on X by a, b, c, . . . ∈ L X . For L-sets a, b, a j ( j ∈ J ), we extend the order relation pointwise, i.e. we write a ⱕ b if for all x ∈ X we have a(x) ⱕ b(x). Also the
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
41
operations in (L , ⱕ , ⊗, ∗) are extended pointwise by ( j∈J a j )(x) = j∈J (a j (x)), ( j∈J a j )(x) = j∈J (a j (x)), (a ⊗ b)(x) = a(x) ⊗ b(x), (a ∗ b)(x) = a(x) ∗ b(x), where x ∈ X . Further we denote for A ⊆ X and ∈ L for all x ∈ A, A (x) = ⊥ else. In particular, A is the characteristic function of A and ⊥ X is the zero function. For notions from category theory we refer to the book [1]. From here on until the end of the paper (unless stated otherwise) we will assume that (L , ⱕ ⊗, ∗) is a pseudobisymmetric enriched cl-premonoid. Let X be a set. A mapping F : L X −→ L with (F1) (F2) (F3) (Fs)
F( X ) = and F(⊥ X ) = ⊥; a ⱕ b implies F(a) ⱕ F(b); F(a) ⊗ F(b) ⱕ F(a ⊗ b); ∗ F(a) ⱕ F( X ∗ a);
(where a, b ∈ L X and ∈ L) is called a stratified L-filter on X [11]. Example. For a point x ∈ X , the point L-filter, [x], is defined by [x](a) = a(x) for a ∈ L X . Then [x] is a stratified L-filter on X. We denote the set of all stratified L-filters on X by F Ls (X ). An order relation on F Ls (X ) can be defined in the natural have F(a) ⱕ G(a). The infimum of a family of stratified L-filters {F j : j ∈ J } way: F ⱕ G if forall a ∈ L X we is then given by ( j∈J F j )(a) = j∈J (F(a)) (see [11]). For a mapping : X −→ Y and F ∈ F Ls (X ) we define ← X ← (F) ∈ F Ls (Y ) by (F)(a) = F( F Ls (Y ) we define the (a)) for←a ∈ L . Here, ←(a) = a ◦ s, see [11]. For G ∈ ← ← inverse image by (G)(a) = {G(b) : (b) ⱕ a}. Then (G) ∈ F L (X ) if and only if (b) = ⊥ X implies s G(b) = ⊥, [12]. As a special case (with the embedding i A s: A −→ X ) we obtain the trace, F A , of F ∈ F L (X ) on A ⊆ X by F A (a) = {F(b) : b| A ⱕ a}. Then F A ∈ F L (A) if and only if b| A = ⊥ A implies F(b) = ⊥, see [12]. We further denote for F ∈ F Ls (A), [F] = i A (F) ∈ F Ls (X ). Then [F](b) = F(b| A ) and [F A ] ⱖ F whenever F A ∈ F Ls (A), [12]. For a set J and G ∈ F Ls (J ) and F j ∈ F Ls (X ) for all j ∈ J we define the stratified L-diagonal filter, G(F (·) ) ∈ F Ls (X ), by G(F (·) )(a) = G(F (·) (a)) with F (·) (a)( j) = F j (a), see [15,16]. For stratified L-filters on the product space X × X , , ∈ F Ls (X × X ), we define −1 −1 X ×X ; • −1 (a) = (a a ∈ L ) with a (x, y) = a(y, x) for (x, y) ∈ X × X and ( f ) ∗ (g) with f ◦ g(x, y) = f (x, z) ∗ g(z, y) for (x, y) ∈ X × X and • ◦ (a) = X ×X
a ∈ L X ×X .
f,g∈L
: f ◦g ⱕ a
z∈X
Then −1 ∈ F Ls (X ×X ) (see [20]). Moreover, ◦ ∈ F Ls (X ×X ) if and only if f ◦g = ⊥ X ×X implies ( f )∗(g) = ⊥. (See [20] for the case of a complete Heyting algebra and [6] for a pseudo-bisymmetric enriched cl-premonoid. The condition for the “existence” of ◦ needs the assumption of pseudo-bisymmetry of the enriched cl-premonoid.) s Let L be a complete Heyting algebra and F, G ∈ F L (X ). We define the product [12], F × G, by F × G(c) = {F( f ) ∧ G(g) : f × g ⱕ c}. Here, f × g(x, y) = f (x) ∧ g(y). For the definition of the product of two stratified L-filters in more general lattice situations see [6]. We further define for F ∈ F Ls (X ) and x ∈ X the stratified L-filter Fx ∈ F Ls (X × X ) by Fx (a) = F(a(·, x)), [6]. If L is a complete Heyting algebra, then Fx = F × [x], see [20]. Lemma 2.2. Let F ∈ F Ls (X ) and denote p1 , p2 : X × X −→ X the first and second projections. Then p1 (Fx ) = F and p2 (Fx ) ⱖ [x]. Proof. We have for a ∈ L X , p1 (Fx )(a) = Fx ( p1← (a)) = F( p1← (a)(·, x)) = F(a(·)) = F(a). Further, p2 (Fx )(a) = Fx ( p2← (a)) = F( p2← (a)(·, x)) = F(a(x) X ) ⱖ a(x) = [x](a), by (Fs). Lemma 2.3. Let G ∈ F Ls (J ) and let = (1 , 2 ) : J −→ X × X . Then for j ∈ J we have (1 (G))2 ( j) = (1 × 2 )(G j ).
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
Proof. We have for b ∈ L X ×X , (1 × 2 )(G j )(b) = G j ((1 × 2 )← (b)) = G((1 × 2 )← (b)(·, j)) = G(b)(1 (·), 2 ( j)) = G(← 1 (b(·, 2 ( j)))) = 1 (G)(b(·, 2 ( j))) = (1 (G))2 ( j) (b). Lemma 2.4. Let L be a complete Heyting algebra, A ⊆ X and F ∈ F Ls (X ) such that F A ∈ F Ls (A). Then [F A × F A ] ⱖ F × F. Proof. Let b ∈ L X ×X . Then F × F(b) = {F(e) ∧ F( f ) : e × f ⱕ b}. If e × f ⱕ b, then for all x, y∈ A, e(x) ∧ f (y) ⱕ b(x, y) and hence we have e| A × f | A ⱕ b| A×A . Moreover, F A (e| A ) ⱖ F(e). Hence F × F(b) ⱕ {F A (e| A ) ∧ F A ( f | A ) : e| A × f | A ⱕ b| A×A } ⱕ F A × F A (b| A×A ) = [F A × F A ](b). We further denote the point L-filter of the point (x, y) ∈ X × X by [(x, y)] ∈ F Ls (X × X ). For a mapping : X −→ Y the product mapping × : X × X −→ Y ×Y is defined by (×)(x, y) = ((x), (y)), where (x, y) ∈ X × X . For a complete Heyting algebra, L, and F, G ∈ F Ls (X ) then we have ( × )(F × G) = (F) × (G). 3. Stratified L-uniform convergence spaces Let X be a set. A mapping : F Ls (X × X ) −→ L is called a stratified L-uniform convergence structure on X [6] ([20] for L a complete Heyting algebra) if (UC1) (UC2) (UC3) (UC4) (UC5)
∀x ∈ X : ([(x, x)]) = ; ⱕ implies () ⱕ (); () ⱕ (−1 ); () ∧ () ⱕ ( ∧ ); () ∗ () ⱕ ( ◦ ) whenever ◦ ∈ F Ls (X × X ).
The pair (X, ) is then called a stratified L-uniform convergence space. A mapping : (X, ) −→ (Y, ) between two stratified L-uniform convergence spaces is called uniformly continuous [6] if for all ∈ F Ls (X × X ) we have () ⱕ (( × )()). The category SL-UCS has as objects the stratified L-uniform convergence spaces and as morphisms the uniformly continuous mappings. It was shown in [6] that SL-UCS is well-fibred and topological over SET and that for ∗ = ⊗ = ∧ (i.e. for the case that L is a complete Heyting algebra) SL-UCS is Cartesian closed [20]. If L = {0, 1}, then stratified L-uniform convergence spaces can be identified with uniform convergence spaces [4] (with the “improved” definition due to Wyler [27]). Initial structures in SL-UCS are given as follows. Let ((X j , j )) j∈J be a family of stratified L-uniform convergence spaces and let X be a set and ( j : X −→ X j ) j∈J be a family of mappings. Then the initial SL-UCS-structure on X is given by
() = j (( j × j )()) j∈J
for ∈ F Ls (X × X ), see [6,20]. In particular, for A ⊆ X and = i A : A −→ X , we obtain the subspace (A, | A ) with | A () = ([]) for ∈ F Ls (A × A). A mapping lim : F Ls (X ) −→ L X is called a stratified L-limit structure on X [6,12,13,25] if (L1) ∀x ∈ X : lim[x]x = ; (L2) F ⱕ G implies lim F ⱕ lim G; (L3) ∀F, G ∈ F Ls (X ) : lim F ∧ lim G ⱕ lim(F ∧ G). The pair (X, lim) is then called a stratified L-limit space. A mapping : (X, lim X ) −→ (Y, limY ) between two stratified L-limit spaces is called continuous if for all F ∈ F Ls (X ) and all x ∈ X we have lim X F(x) ⱕ limY (F)((x)). The category SL-LIM with stratified L-limit spaces as objects and continuous mappings as morphisms is well-fibred and topological over SET and, in case ∗ = ⊗ = ∧, Cartesian closed [13,25]. Initial structures in SL-LIM are given as follows. Let ((X j , lim j )) j∈J be a family of stratified L-limit spaces and let X be a set and ( j : X −→ X j ) j∈J be a family of mappings. Then the initial SL-LIM-structure on X is given by
lim F(x) = lim j j (F)( j (x)) j∈J
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
43
for F ∈ F Ls (X ) and x ∈ X , see [12,13,25]. If X = X 1 × X 2 and 1 = p1 : X 1 × X 2 −→ X 1 and 2 = p2 : X 1 × X 2 − → X 2 are the projections, then we denote the initial limit structure by lim1 × lim2 . Lemma 3.1. Let (X, lim X ), (Y, limY ) ∈ |S L-L I M| and let : (X, lim X ) −→ (Y, limY ) be continuous. Then × : (X × X, lim X × lim X ) −→ (Y × Y, limY × limY ) is continuous. Proof. We denote the projections from X × X to X by p1 and p2 and the projections from Y × Y to Y by q1 and q2 . Then q1 ◦ ( × ) = ◦ p1 and q2 ◦ ( × ) = ◦ p2 . We have for ∈ F Ls (X × X ), lim X × lim X (x, y) = lim X p1 ()(x) ∧ lim X p2 ()(y) ⱕ limY ◦ p1 ()((x)) ∧ limY ◦ p2 ()((y)) = limY q1 ◦ ( × )()(q1 ◦ ( × )(x, y)) ∧ limY q2 ◦ ( × )()(q2 ◦ ( × )(x, y)) = limY × limY ( × )()((x), (y)). We can define a forgetful functor from SL-UCS to SL-LIM by ⎧ ⎪ ⎨ S L-U C S −→ S L-L I M (X, lim( )) F : (X, ) ⎪ ⎩ , where lim( )F(x) = (Fx ). This functor preserves initial structures, [20]. In particular for subspaces of (X, ) we have ( A, lim( | A )) = (A, lim( )| A ). We will later need the following result. Lemma 3.2. Let ((X j , j )) j∈J be a family of spaces in SL-UCS, let X be a set and let ( j : X −→ X j ) j∈J be a family of mappings. If (X, ) is the initial construction in SL-UCS, then for ∈ F Ls (X × X ) and (u, v) ∈ X × X we have
lim( j ) × lim( j )( j × j )()(( j × j )(u, v)). lim( ) × lim( )(u, v) ⱕ j∈J j
Proof. We denote the projections from X × X to X by p1 and p2 and the projections from X j × X j to X j by p1 and j j j p2 . Then we have p1 ◦ ( j × j ) = j ◦ p1 and p2 ◦ ( j × j ) = j ◦ p2 . Hence we have lim( ) × lim( )(u, v) = lim( ) p1 ()(u) ∧ lim( ) p2 ()(v)
= lim( j ) j ◦ p1 ()( j (u)) ∧ lim( i )i ◦ p2 ()(i (v)) j∈J
=
i∈J
j
j
lim( j ) p1 ◦ ( j × j )()( p1 ◦ ( j × j )(u, v))
j∈J
∧ ⱕ
lim( i ) p2i ◦ (i × i )()( p2i ◦ (i × i )(u, v))
i∈J j
j
j
j
[lim( j ) p1 ◦ ( j × j )()( p1 ◦ ( j × j )(u, v))
j∈J
∧ lim( j ) p2 ◦ ( j × j )()( p2 ◦ ( j × j )(u, v))]
lim( j ) × lim( j )( j × j )()(( j × j )(u, v)). = j∈J
In [14,15,25] we defined the following diagonal conditions in SL-LIM. (LK) ∀G ∈ F Ls (X ), ∀F y ∈ F Ls (X ) (y ∈ X ), ∀x ∈ X :
lim F y (y) ⱕ lim G(F (·) )(x). lim G(x) ∗ y∈X
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
(LF) ∀J, ∀ : J −→ X, ∀G ∈ F Ls (J ), ∀F j ∈ F Ls (X )( j ∈ J ), ∀x ∈ X :
lim F j (( j)) ⱕ lim G(F (·) )(x). lim (G)(x) ∗ j∈J
The condition (LK) generalizes a diagonal condition for limit spaces due to Kowalsky [22] and (LF) generalizes a diagonal condition attributed to Fischer, see [5], from the “classical case”, L = {0, 1}, to the general lattice-valued case. For more information on these conditions see [14,15,25]. A space (X, lim) ∈ |S L-L I M| is called regular [16] if it satisfies the following “dual axiom” of the axiom (LF): (LR) ∀J, ∀ : J −→ X, ∀G ∈ F Ls (J ), ∀F j ∈ F Ls (X )( j ∈ J ), ∀x ∈ X :
lim F j (( j)) ⱕ lim (G)(x). lim G(F(·) )(x) ∗ j∈J
If L is a complete Heyting algebra, then regularity can be characterized by certain closures of stratified L-filters, see [2] (and for L a complete Boolean algebra, see [16]). For F ∈ F Ls (X ) and ∈ L we define J = {(G, x) : lim G(x) ⱖ } and with this the -closure of F by F (a) = {F(b) ∧ : G(b) ∧ ⱕ a(x)∀(G, x) ∈ J } (a ∈ L X ).
Then (X, lim) is regular if and only if lim F (x) ⱖ ∧ whenever lim F(x) ⱖ . Note that in [2] the definition of F was stated with an additional condition, F (a) = {F(b) ∧ : b ∧ X ⱕ a, G(b) ∧ ⱕ a(x)∀(G, x) ∈ J }. However, if G(b) ∧ ⱕ a(x)∀(G, x) ∈ J then always b ∧ X ⱕ a, because ([x], x) ∈ J for all x ∈ X . 4. Kowalsky and Fischer type diagonal conditions in S L-U C S We first present the following adaptation of Kowalsky’s diagonal condition for a space (X, ) ∈ |S L-U C S|. We need to introduce the following notation. For ∈ F Ls (X × X ) and a family of stratified L-filters on X × X indexed by X × X , {(u,v) : (u, v) ∈ X × X } ⊆ F Ls (X × X ), we define ((·,−) ) ∈ F Ls (X × X ) by ((·,−) )(a) = ((·,−) (a)) (a ∈ L X ×X ), with (·,−) (a)(u, v) = (u,v) (a) for (u, v) ∈ X × X . (LUK) ∀ ∈ F Ls (X × X ), ∀(u,v) ∈ F Ls (X × X ), ((u, v) ∈ X × X ) :
lim( ) × lim( )(u,v) (u, v) ⱕ (((·,−) )). () ∗ (u,v)∈X ×X
Gähler [7] called a classical uniform convergence structure which satisfies (LUK) a diagonal uniform limit structure (“limesuniforme Diagonalstruktur”) . Hence we shall call (X, ) ∈ |S L-U C S| a stratified Kowalsky-diagonal L-uniform convergence space if (LUK) is satisfied. The following result seems to be new also in the classical case. Lemma 4.1. Let (X j , j ) ∈ |S L-U C S| satisfy (LUK) for all j ∈ J and let X be a set and for all j ∈ J , let j : X −→ X j be injective. Then the initial construction (X, ) satisfies (LUK). Proof. Let ∈ F Ls (X × X ) and for (u, v) ∈ X × X let (u,v) ∈ F Ls (X × X ). For j ∈ J and (x j , y j ) ∈ ( j × j )(X × X ) (x j ,y j )
we define j
= ( j × j )((u,v) ) with the unique (u, v) = ( j × j )−1 (x j , y j ). For (x j , y j ) ∈ / ( j × j )(X × X )
we define
= [(x j , y j )]. Then
(x ,y ) j j j
(x ,y )
lim( j ) × lim( j ) j j j (x j , y j ) lim( j ) × lim( j )( j × j )((u,v) )(( j × j )(u, v)) if (x j , y j ) ∈ ( j × j )(X × X ), = lim( j ) × lim( j )[(x j , y j )](x j , y j ) = otherwise.
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
45
Hence, with Lemma 3.2 and (LUK) for the spaces (X j , j ), we conclude
lim( ) × lim( )(u,v) (u, v) () ∗ (u,v)∈X ×X
ⱕ
j∈J
ⱕ
⎡
⎣ j (( j × j )()) ∗
ⱕ
⎤ lim( j ) × lim( j )( j × j )((u,v) )( j (u), j (v))⎦
(u,v)∈X ×X
⎡
⎤
⎣ j (( j × j )()) ∗
(x ,y ) lim( j ) × lim( j ) j j j (x j ,
y j )⎦
(x j ,y j )∈X j ×X j
j∈J
(·,−)
j (( j × j )()( j
)).
j∈J (x j ,y j )
Now we note that for (z, w) ∈ X × X we have, by the definition of j ( j × j )← [ j
(·,−)
(·,−)
(a)](z, w) = [ j
,
( j (z), j (w))
(a)]( j (z), j (w)) = j
(a)
= ( j × j )((z,w) )(a) = ( j × j )((·,−) )(a)(z, w). Hence we have for a ∈ L X j ×X j (·,−)
[( j × j )()]( j
)(a) = (( j × j )← [(·,−) )(a)]) = ([( j × j )((·,−) )](a))
= ((·,−) (( j × j )← (a))) = ((·,−) )(( j × j )← (a)) = ( j × j )(((·,−) ))(a), (·,−)
i.e. we have [( j × j )()]( j () ∗
) = ( j × j )(((·,−) )). Therefore
lim( ) × lim( )(u,v) (u, v) ⱕ
(u,v)∈X ×X
j (( j × j )(((·,−) ))) = (((·,−) )).
j∈J
In particular, a subspace of a stratified Kowalsky-diagonal L-uniform convergence space satisfies (LUK). Lemma 4.2. Let (X, ) ∈ |S L-U C S| satisfy the axiom (LUK). Then (X, lim( )) satisfies (LK). Proof. Let G ∈ F Ls (X ) and for each y ∈ X , let F y ∈ F Ls (X ) and let x ∈ X . Define = Gx and for (u, v) ∈ X × X define (u,v) = (F u )v . Then () = (Gx ) = lim( )G(x) and lim( ) × lim( )(u,v) (u, v) = lim( ) p1 ((u,v) )(u) ∧ lim( ) p2 ((u,v) )(v) ⱖ lim( )F u (u) ∧ lim( )[v](v) = lim( )F u (u). Moreover, we have for a ∈ L X ×X ((·,−) )(a) = Gx ((F (·) )(−) )(a) = Gx ((F (·) )(−) (a)) = G((F (·) )x (a)) = G(F (·) (a(∗, x))) = (G(F (·) ))(a(∗, x)) = (G(F (·) ))x (a). Hence we obtain lim( )G(x) ∗
lim( )F y (y) ⱕ () ∗
lim( ) × lim( )(u,v) (u, v)
(u,v)∈X ×X
y∈X
ⱕ ((
(·,−)
) = ((G(F (·) ))x ) = lim( )G(F (·) )(x),
i.e. (LK) is true. In the following we present an adaptation of Fischer’s axiom for a space (X, ) ∈ |S L-U C S|. This axiom has not been studied before in the classical case, L = {0, 1}.
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
(LUF) ∀J, ∀ : J −→ X × X, ∀ ∈ F Ls (J ), ∀ j ∈ F Ls (X × X ) ( j ∈ J ) :
lim( ) × lim( ) j (( j)) ⱕ (((·) )). (()) ∗ j∈J
We shall call a space (X, ) ∈ |S L-U C S| that satisfies (LUF) a stratified Fischer-diagonal L-uniform convergence space. Clearly, for J = X × X and = id X ×X we see that (LUF) implies (LUK). The axiom (LUF) is preserved under initial constructions. Lemma 4.3. Let (X j , j ) ∈ |S L-U C S| satisfy (LUF) for all j ∈ J and let X be a set and for all j ∈ J , let j : X −→ X j be mappings. Then the initial construction (X, ) satisfies (LUF). Proof. Let M be a set, : M −→ X × X , ∈ F Ls (M) and let for m ∈ M, m ∈ F Ls (X × X ). We define (·) j = ( j × j ) ◦ and mj = ( j × j )(m ). Then ( j ) = (( j × j )((·) )) = ( j × j )(((·) )) (see [15]). Hence we conclude
(()) ∗ lim( ) × lim( )m ((m)) ⱕ
m∈M
j (( j × j )(())) ∗
j∈J
=
j ( j ()) ∗
j∈J
ⱕ
lim( j ) × lim( j )( j × j )( )(( j × j )((m)))
m∈M
m
lim( j ) × lim( j )mj ( j (m))
m∈M (·) j (( j ))
j∈J
=
j (( j × j )(((·) ))) = (((·) )).
j∈J
In particular, products and subspaces of stratified Fischer-diagonal L-uniform convergence spaces satisfy (LUF). Because SL-UCS is a topological category and, according to Lemma 4.3, the subcategory of spaces that satisfy (LUF) is initially closed in SL-UCS, it is reflective in SL-UCS, see [1]. Lemma 4.4. Let (X, ) ∈ |S L-U C S| satisfy the axiom (LUF). Then (X, lim( )) satisfies (LF). Proof. Let J be a set, : J −→ X , G ∈ F Ls (J ) and for each j ∈ J , F j ∈ F Ls (X ) and x ∈ X . We define J = J × {x} and : J −→ X × X by ( j, x) = (( j), x). Further we define = Gx ∈ F Ls ( J) by Gx (a) = G(a(·, x)) for a ∈ L J ( j,x) j X ×X and = (F )x . We then have for a ∈ L and j ∈ J , ← (a(·, x))( j) = a(·, x)(( j)) = a(( j), x) = a(( j, x)) = ← (a)( j, x) and hence (G)x (a) = (G)(a(·, x)) = G(← (a(·, x))) = G(← (a)(·, x)) = Gx (← (a)) = (Gx )(a), i.e. we have (G)x = (). Hence lim( )(G)(x) = (((G))x ) = (()). Similarly as in the proof of Lemma 4.2 we find lim( ) × lim( ) j (( j)) ⱖ lim( )F j (( j)). Hence
lim( )F j (( j)) lim( )(G)(x) ∗ j∈J
ⱕ (()) ∗
j∈J
ⱕ (((·) )).
lim( ) × lim( ) j (( j))
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
47
Because, for a ∈ L X ×X , we have ((·) )(a) = Gx ((·) (a)) = G((·,x) (a)) = G((F (·) )x (a)) = G(F (·) (a(∗, x))) = G(F (·) )(a(∗, x)) = (G(F (·) ))x (a) we conclude (((·) )) = ((G(F (·) ))x ) = lim( )G(F (·) )(x) and (X, lim( )) satisfies (LF). We are finally going to characterize the axiom (LUF) by a neighbourhood condition. We define for ∈ L and (x, y) ∈ X × X the -level neighbourhood L-filter of (x, y) with respect to lim( ) × lim( ) by
(x,y) = . U lim( )×lim( )(x,y) ⱖ (x,y)
Clearly, U of by
(x,y)
∈ F Ls (X × X ) and U
ⱕ [(x, y)]. For ∈ F Ls (X × X ) we define the -level neighbourhood L-filter
(·)
U () = (U ). (x,y)
Then U ([(x, y)]) = U and U () ⱕ . For a similar construction for a space (X, lim) ∈ |S L-L I M| see [19]. We say that a space (X, ) ∈ |S L-U C S| satisfies the uniform neighbourhood condition if the axiom (LUG) ∀, ∈ L , ∀ ∈ F Ls (X × X ) : () ⱖ ⇒ (U ()) ⱖ ∗ is satisfied. For the classical case, L = {0, 1}, this condition goes back to the work of W. Gähler [7]. Theorem 4.5. Let (X, ) ∈ |S L-U C S|. Then (X, ) satisfies (LUF) if and only if it satisfies (LUG). Proof. Let first (X, ) satisfy (LUF) and let ∈ F Ls (X × X ) and ⱕ (). We define J = {(, (x, y)) : lim( ) × lim( )(x, y) ⱖ } and : J −→ X × X, (, (x, y)) (x, y) and for (, (x, y)) ∈ J we define (,(x,y)) = . Because lim( ) × lim( )[(x, y)](x, y) = ⱖ , the mapping is surjective. Hence K = ← () ∈ F Ls (J ) and (K) = . Furthermore,
lim( ) × lim( )(,(x,y)) ((, (x, y))) = lim( ) × lim( )(x, y) ⱖ . (,(x,y))∈J
(,(x,y))∈J
By (LUF) then ∗ ⱕ (K((·) )). We show that K((·) ) ⱕ U (). Let a ∈ L X ×X . Then K((·) )(a) = ← ()((·) (a)) = {(b) : ← (b) ⱕ (·) (a)}. If ← (b) ⱕ (·) (a), then for all (, (x, y)) ∈ J we have b(x, y) = ← (b)((, (x, y))) ⱕ ( ,(x,y)) (a) = (a). Hence
(x,y) b(x, y) ⱕ (a) = U (a) lim( )×lim( )(x,y) ⱖ
and we conclude K((·) )(a) ⱕ
(·)
(·)
{(b) : b ⱕ U (a)} ⱕ (U (a)) = U ()(a).
Hence the condition (LUG) is satisfied. Let now (X, ) satisfy (LUG). Let J be a set, : J −→ X × X , ∈ F Ls (J ) and for each j ∈ J let j ∈ F Ls (X × X ). ( j) We put = (()) and = j∈J lim( ) × lim( ) j (( j)). Then for all j ∈ J we have that U ⱕ j . By (LUG) moreover ∗ ⱕ (U (())). We show that U (()) ⱕ ((·) ). Let a ∈ L X ×X . Then (·)
(·)
U (())(a) = ()(U (a)) = (U
(a)) ⱕ ((·) (a)) = ((·) )(a).
Hence ∗ ⱕ (((·) )) and (LUF) is true. Remark 4.6. In [19] we introduced a similar condition, (LG∗ ), in the category SL-LIM: (LG∗ ) ∀, ∈ L , ∀F ∈ F Ls (X ), ∀x ∈ X : lim F(x) ⱖ ⇒ lim U (F)(x) ⱖ ∗ .
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
(·) Here, Ux = lim F (x) ⱖ F and U (F) = F(U ). We showed that this condition is equivalent to the axiom (LF). From Lemmas 4.4 and Theorem 4.5 then we conclude that if a space (X, ) ∈ |S L-U C S| satisfies (LUG), then (X, lim( )) satisfies (LG∗ ). 5. Uniform regularity A space (X, ) ∈ |S L-U C S| is called uniformly regular if it satisfies the following “dual axiom” of the axiom (LUF): ∀J, ∀ : J −→ X × X, ∀ ∈ F Ls (J ), ∀ j ∈ F Ls (X × X ) ( j ∈ J ) :
lim( ) × lim( ) j (( j)) ⱕ (()). (((·) )) ∗
(LUR)
j∈J
The axiom (LUR) generalizes the definition of uniform regularity from the classical case, L = {0, 1} as studied by Gähler [7], to the lattice-valued case. Lemma 5.1. Let (X j , j ) ∈ |S L-U C S| satisfy (LUR) for all j ∈ J and let X be a set and for all j ∈ J , let j : X −→ X j be mappings. Then the initial construction (X, ) satisfies (LUR). Proof. This proof is similar to the proof of Lemma 4.3 and is not presented. In particular, products and subspaces of uniformly regular L-uniform convergence spaces satisfy (LUR). Because SL-UCS is a topological category and, according to Lemma 5.1, the subcategory of spaces that satisfy (LUR) is initially closed in SL-UCS, it is reflective in SL-UCS, see [1]. Lemma 5.2. Let (X, ) ∈ |S L-U C S| satisfy the axiom (LUR). Then (X, lim( )) satisfies (LR). Proof. Also this proof uses methods similar to the methods of the proof of Lemma 4.4 and is not presented. We are now going to characterize uniform regularity by closures of stratified L-filters. We will restrict to the case that L is a complete Heyting algebra. The constructions and proof are simple extensions of results in [2,16]. For ∈ F Ls (X × X ) and ∈ L we define J = {(, (x, y)) : lim( ) × lim( )(x, y) ⱖ } and with this the -closure of by (a) = {(b) ∧ : (b) ∧ ⱕ a(x, y)∀(, (x, y)) ∈ J },
where a ∈ L X ×X . Then it is not difficult to show that ∈ F Ls (X × X ) and ⱕ . Moreover for ⱕ we have
ⱕ . Theorem 5.3. Let L be a complete Heyting algebra and let (X, ) ∈ |S L-U C S|. Then (X, ) is uniformly regular if and only if for all , ∈ L and all ∈ F Ls (X × X ) we have ( ) ⱖ ∧ whenever () ⱖ . Proof. Let (X, ) satisfy the axiom (LUR) and let () ⱖ . We define again J = {(, (x, y)) : ∈ F Ls (X × X ), lim( ) × lim( )(x, y) ⱖ }. For j = (, (x, y)) ∈ J we define j = and we define the mapping : J −→ X × X by ((, (x, y)) = (x, y). Then lim( ) × lim( ) j (( j)) = lim( ) × lim( )(x, y) ⱖ , i.e. ( j , (( j)) ∈ J whenever j ∈ J . We define for a ∈ L J , S(a) = (( f ) ∧ ). ( f )∧ ⱕ a( j) ∀ j=(,(x,y))∈J
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
49
Then it is not difficult to show that S ∈ F Ls (J ). Moreover, ⱕ S((·) ). To see this, let a ∈ L X ×X . Then
S((·) )(a) =
(( f ) ∧ ) =
(( f ) ∧ ) ⱖ (a) ∧ = (a).
( f )∧ ⱕ (a) ∀ j∈J
( f )∧ ⱕ (a) ∀ j∈J j
With the axiom (LUR) we conclude
((S)) ⱖ (S((·) )) ∧ lim( ) × lim( ) j (( j)) ⱖ () ∧ ⱖ ∧ . j∈J
We therefore only have to show that (S) ⱕ . Let a ∈ L X ×X . Then ← (a)((, (x, y))) = a(((, (x, y))) = a(x, y) and hence (( f ) ∧ ) (S)(a) = S(← (a)) = ( f )∧ ⱕ ← (a)((,(x,y))) ∀(,(x,y))∈J
=
(( f ) ∧ ).
( f )∧ ⱕ a(x,y) ∀(,(x,y))∈J
Hence (S)(a) ⱕ (a) and the condition of the Theorem is satisfied. Conversely, we assume that the condition of the Theorem is satisfied. Let J be a set, : J −→ X × X , ∈ F Ls (J ) and for each j ∈ J let j ∈ F Ls (X × X ). Define = j∈J lim( ) × lim( )( j)(( j)) and = (((·) )). Then
∧ ⱕ (((·) )) ). We will show that ((·) )) ⱕ (). Let a ∈ L X ×X . Then by definition
((·) ) (a) =
{((·) )( f ) ∧ : ( f ) ∧ ⱕ a(x, y) ∀(, (x, y)) ∈ J }.
We fix f ∈ L X ×X and ∈ L such that ( f ) ∧ ⱕ a(x, y) for all (, (x, y)) ∈ J . Because ( j , ( j)) ∈ J , then especially j ( f ) ∧ ⱕ a(( j)) = ← (a)( j) and it follows, by (Fs), that ((·) )( f ) ∧ ⱕ ((·) ( f ) ∧ ) ⱕ (← (a)(·)) = ()(a). Hence ∧ ⱕ (()) and the proof is complete. Corollary 5.4. Let L be a complete Heyting algebra and let (X, ) ∈ |S L-U C S|. Then (X, ) is uniformly regular if and only if for all ∈ L and all ∈ F Ls (X × X ) we have ( ) ⱖ whenever () ⱖ . If the lattice L is given by a complete Boolean algebra, then it was shown in [18] that a space (X, ) ∈ |S L-U C S| induces a stratified L-Cauchy space (X, C ) by defining, for F ∈ F Ls (X ), C (F) = (F × F). This means that (X, C ) satisfies the axioms (LC1) C ([x]) = for all x ∈ X , (LC2) F ⱕ G implies C (F) ⱕ C (G) and (LC3) C (F ∧ G) = C (F) ∧ C (G) whenever F ∨ G ∈ F Ls (X ). We will in the remainder of this section assume that L is a complete Boolean algebra. Following [3] we call (X, C ) regular if
(LCR) ∀ ∈ L ∀F ∈ F Ls (X ) : ⱕ C (F) ⇒ ⱕ C (F ). Here, F (a) = {F(b) : lim(C )G(x) ⱖ ⇒ G(b) ⱕ a(x)}, see [16], where lim(C )F(x) = C (F ∧ [x]). We note that lim(C ) = lim( ) [18]. In order to show that a uniformly regular stratified L-uniform convergence space induces a regular stratified L-Cauchy space, we introduce a new notation. We define for b ∈ L X and ∈ L the set L = {(G, x) : lim( )G(x) ⱖ } and b (x) = G(b). (G ,x)∈L
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
It is then not difficult to prove that, for a ∈ L X , we have that F (a) = {F(b) : b ⱕ a}. Similarly, for b ∈ L X ×X and ∈ L we denote J = {(, (x, y)) : lim( ) × lim( )(x, y) ⱖ } and b (x, y) = (b). (,(x,y))∈J
Then, for a ∈ L X ×X , we have that (a) = {(b) : b ⱕ a}. Note that we do not distinguish the notation for a ∈ L X and a ∈ L X ×X . The corresponding interpretation should be clear from the context.
Lemma 5.5. Let L be a complete Boolean algebra, (X, ) ∈ |S L-U C S|, ∈ L and let f 1 , f 2 ∈ L X . Then f 1 × f2 ⱕ f1 × f2 . Proof. Let (x, y) ∈ X × X . We note that (G, x), (H, y) ∈ L implies (G × H, (x, y)) ∈ J . Further, for = G × H we have that p1 () ⱖ G and p2 () ⱖ H. Hence we conclude
f 1 × f 2 (x, y) = f 1 (x) ∧ f 2 (y) G( f 1 ) ∧ = (G ,x)∈L
H( f 2 )
(H,y)∈L
ⱕ
(G( f 1 ) ∧ H( f 2 ))
(G ×H.(x,y))∈J
ⱕ
( p1 ()( f 1 ) ∧ p2 ()( f 2 ))
(,(x,y))∈J
=
(( p1← ( f 1 )) ∧ ( p2← ( f 2 )))
(,(x,y))∈J
=
( p1← ( f 1 ) ∧ p2← ( f 2 ))
(,(x,y))∈J
=
( f 1 × f 2 )
(,(x,y))∈J
= f 1 × f 2 (x, y).
Lemma 5.6. Let L be a complete Boolean algebra, (X, ) ∈ |S L-U C S|, ∈ L and let F ∈ F Ls (X ). Then F × F ⱕ F ×F . Proof. Let a ∈ L X ×X . Then, using Lemma 5.5, we obtain F × F (a) = F × F(b)
b ⱕa
=
(F( f 1 ) ∧ F( f 2 ))
b ⱕ a f1 × f2 ⱕ b
ⱕ
(F( f 1 ) ∧ F( f 2 ))
f1 × f2 ⱕ a
ⱕ
(F( f 1 ) ∧ F( f 2 )).
f1 × f2 ⱕ a
Now we note that f 1 × f 2 ⱕ a if and only if there are b, c ∈ L X such that b × c ⱕ a and f 1 ⱕ b and f 2 ⱕ c. Hence, using the frame law, we conclude
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
F × F (a) ⱕ
(F( f 1 ) ∧ F( f 2 ))
b×c ⱕ a f 1 ⱕ b, f 2 ⱕ c
=
⎛ ⎝
=
F( f 1 ) ∧
b×c ⱕ a
⎞ F( f 2 )⎠
f1 ⱕ b
51
f2 ⱕ c
(F (b) ∧ F (c))
b×c ⱕ a
= F × F (a).
Lemma 5.7. Let L be a complete Boolean algebra. If (X, ) ∈ |S L-U C S| satisfies the axiom (LUR), then (X, C ) satisfies (LCR). Proof. Let F ∈ F Ls (X ) and let ∈ L and ⱕ C (F). Then ⱕ (F × F). With Corollary 5.4 we conclude ⱕ (F × F ) and hence, by Lemma 5.6, ⱕ (F × F ) = C (F ). 6. An extension theorem for uniformly continuous mappings In this section we restrict the lattice context to complete Heyting algebras. In [16] we proved an Extension Theorem for continuous mappings. We first recall some definitions and notation. For (X, lim X ), (Y, limY ) ∈ |S L-L I M| and A ⊆ X and a mapping : A −→ Y we define, for ∈ L and x ∈ X , H A (x) = {F ∈ F Ls (X ) : F A ∈ F Ls (A) and lim X F(x) ⱖ }, FA (x) = {y ∈ Y : limY (F A )(y) ⱖ for all F ∈ H A (x)}. A subset A ⊆ X is called dense in (X, lim X ) if from x ∈ X it follows that H A (x) ∅. Further we call (X, lim X ) a T2-space if for all F ∈ F Ls (X ), x = y whenever lim X F(x) = lim X F(y) = . We will also need a stronger separation axiom. We will call (X, lim) a strong T2-space if for all F ∈ F Ls (X ), x = y whenever lim X F(x) = and lim X F(y) > ⊥. Clearly every strong T2-space is a T2-space. Theorem 6.1 (Jäger [16]). Let L be a complete Heyting algebra and let (X, lim X ) ∈ |S L-L I M| satisfy the axiom (LK) and let (Y, limY ) ∈ |S L-L I M| be a regular T2-space. Further let A ⊆ X be dense in (X, lim X ). Then a continuous mapping : (A, lim X | A ) −→ (Y, limY )has a unique continuous extension, : (X, lim X ) −→ (Y, limY ), such that | A = , if and only if for every x ∈ X , ∈L FA (x) ∅. For (X, ) ∈ |S L-U C S| we call A ⊆ X dense if A is dense in (X, lim( )). Likewise, we call (X, ) a strong T2-space, whenever (X, lim( )) is a strong T2-space. We further call (X, ) complete if for every F ∈ F Ls (X ) there is x ∈ X such that (F × F) ⱕ lim( )F(x). (See in this regard [18].) The following Theorem generalizes a result (4.12.6) of Gähler in [7], to the lattice-valued case. Theorem 6.2. Let L be a complete Heyting algebra and let (X, X ), (Y, Y ) ∈ |S L-U C S|. Further, let (X, X ) satisfy the axiom (LUG) and let (Y, Y ) be a complete, uniformly regular, strong T2-space. If A ⊆ X is dense in (X, lim( X )) and : (A, X | A ) −→ (Y, Y ) is uniformly continuous, then there is a unique uniformly continuous extension : (X, X ) −→ (Y, Y ), such that | A = . Proof. (X, lim( X )) satisfies (LF) and therefore also (LK). Further, (Y, lim( Y )) is a regular T2-space. Let x ∈ X . We show that ∈L FA (x) ∅. Because A is dense in (X, lim( X )), there is F ∈ F Ls (X ) such that F A ∈ F Ls (A) and lim( X )F(x) = . Then, using (UC3), we conclude = X (F × [x]) = ([x] × F) and, with (UC5), therefore also (see [18], Lemma 5.1) = X ((F × [x]) ◦ ([x] × F)) = X (F × F). Then also X | A (F A × F A ) = X ([F A × F A ]) ⱖ X (F × F) = . Because is uniformly continuous then Y (( × )(F A × F A )) = Y ((F A ) × (F A )) = . As (Y, Y ) is complete, there is y0 ∈ Y such that lim( Y )(F A )(y0 ) = . We show that y0 ∈ ∈L FA (x). Let G ∈ H A (x). We may assume that > ⊥. Then G A ∈ F Ls (A) and lim( X )G(x) ⱖ . Then also lim( )(F ∧ G)(x) ⱖ
52
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
and hence ((F ∧ G) × (F × G)) ⱖ . We conclude that (F ∧ G) A ∈ F Ls (A) and X | A ((F ∧ G) A × (F ∧ G) A ) ⱖ . Again, by the uniform continuity of , then Y (((F ∧ G) A ) × ((F ∧ G) A )) ⱖ . Hence, by completeness of (Y, Y ) there is y1 ∈ Y such that lim( Y )((F ∧ G) A )(y1 ) ⱖ . Therefore also lim( Y )(F A )(y1 ) ⱖ and from the strong T2 property we see that y0 = y1 . Hence y0 ∈ FA (x). We can therefore apply Theorem 6.1 and find a continuous extension : (X, lim( X )) −→ (Y, lim( Y )) with | A = . We show that : (X, X ) −→ (Y, Y ) is uniformly continuous. Let ∈ F Ls (X × X ) and let X () = . By (LUG) then also ⱕ X (U ()). Then U () A×A ∈ F Ls (A × A). To see this, let b| A×A = ⊥ A×A . Then, for (x, y) ∈ X × X ,
(x,y) U (b) ⱕ (b) = ⊥, lim( X )×lim( X )(x,y) ⱖ ,( A×A )=
because from the denseness of A we find F, G ∈ F Ls (X ) such that lim( X )[F A ](x) ⱖ and lim( X )[G A ](y) ⱖ . It is not difficult to see that = [F A ] × [G A ] then satisfies lim( X ) × lim( X )(x, y) ⱖ and ( A×A ) = . Hence (·,−) U ()(b) = (U (b)) = (⊥ X ×X ) = ⊥. Then X | A (U () A×A ) = X ([U () A×A ]) ⱖ X (U ()) ⱖ . From the uniform continuity of and because (Y, Y ) is uniformly regular we conclude
ⱕ Y (( × )(U () A×A )) ⱕ Y (( × )(U () A×A ) ).
We finally show that ( × )(U () A×A ) ⱕ ( × )(). Let b ∈ L Y ×Y . Then, with JY = {(, (y1 , y2 )) : lim( Y ) × lim( Y )(y1 , y2 ) ⱖ },
( × )(U () A×A ) (b) (·,−) = {(U ([( × )← (c)]∗ )) ∧ : (c) ∧ ⱕ b(y1 , y2 )∀(, (y1 , y2 )) ∈ JY }, where we denote ←
∗
[( × ) (c)] (x1 , x2 ) =
( × )← (c)(x1 , x2 ) if (x1 , x2 ) ∈ A × A,
otherwise.
It is easy to show that, with a similar notation, [( × )← (c)| A×A ]∗ = [( × )← (c)]∗ . We fix now c ∈ L Y ×Y and ∈ L such that (c) ∧ ⱕ b(y1 , y2 ) for all (, (y1 , y2 )) ∈ JY . Then (x1 ,x2 )
U
=
([( × )← (c)]∗ )
([( × )← (c)| A×A ]∗ )
lim( X )×lim( X )(x1 ,x2 ) ⱖ
ⱕ
[ ]([( × )← (c)| A×A ]∗ )
lim( X )×lim( X )[ ](x1 ,x2 ) ⱖ , ∈F Ls (A×A)
=
(( × )← (c)| A×A )
lim( X )×lim( X )[ ](x1 ,x2 ) ⱖ , ∈F Ls (A×A)
=
[ ](( × )← (c))
lim( X )×lim( X )[ ](x1 ,x2 ) ⱖ , ∈F Ls (A×A)
=
( × )([ ])(c).
lim( X )×lim( X )[ ](x1 ,x2 ) ⱖ , ∈F Ls (A×A)
If lim( X ) × lim( X )[ ](x1 , x2 ) ⱖ , then by continuity of × with respect to lim( X ) × lim( X ) and lim( Y ) × lim( Y ) we have (( × )([ ]), ((x1 ), (x2 ))) ∈ JY . By choice of c and then ( × )([ ])(c) ∧ ⱕ ( × (x ,x ) )← (b)(x1 , x2 ). Hence U 1 2 ([( × )← (c)]∗ ) ∧ ⱕ ( × )← (b)(x1 , x2 ) and we obtain with the stratification condition (Fs)
(·,−)
( f × )(U () A×A ) (b) ⱕ (U This completes the proof.
([( × )← (c)]∗ ∧ ) ⱕ (( × )← (b)) = ( × )()(b).
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
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7. Conclusions We defined and studied several diagonal conditions for lattice-valued uniform convergence spaces. We especially showed that a Fischer-type diagonal condition is equivalent to a neighbourhood condition which was studied for L = {0, 1} by Gähler [7]. Uniform regularity was defined and characterized by closures of stratified L-filters. All the studied notions were finally used to obtain an extension theorem for uniformly continuous mappings. In the classical case, it was shown in [7] that a uniform space satisfies all these diagonal conditions, but that, however, the diagonal conditions do not characterize the uniform spaces among the uniform convergence spaces. This is in contrast to the situation in the category LIM of limit spaces, where e.g. Fischer’s diagonal condition characterizes topological limit spaces among the limit spaces. It is at present not known if lattice-valued uniform spaces [8,9] are uniformly regular or if they satisfy the axiom (LUF). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
J. Adámek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1989. H. Boustique, G. Richardson, A note on regularity, Fuzzy Set. Syst. 162 (2011) 64–66. H. Boustique, G. Richardson, Regularity: lattice-valued Cauchy spaces, Fuzzy Set. Syst. 190 (2012) 94–104. C.H. Cook, H.R. Fischer, Uniform convergence structures, Math. Ann. 173 (1967) 290–306. C.H. Cook, H.R. Fischer, Regular convergence spaces, Math. Ann. 174 (1967) 1–7. A. Craig, G. Jäger, A common framework for lattice-valued uniform spaces and probabilistic uniform limit spaces, Fuzzy Set. Syst. 160 (2009) 1177–1203. W. Gähler, Grundstrukturen der Analysis I, Birkhäuser, Basel and Stuttgart, 1977. J. Gutiérrez García, A Unified Approach to the Concept of Fuzzy L-uniform Space, Thesis, Universidad del Pais Vasco, Bilbo, Spain, 2000. J. Gutiérrez García, M.A. de Prada Vicente, A.P. Šostak, A unified approach to the concept of fuzzy L-uniform space, in: S.E. Rodabaugh, E.P. Klement (Eds.), Topological and algebraic structures in fuzzy sets, Kluwer, Dordrecht, 2003, pp. 81–114. U. Höhle, Commutative residuated l-monoids, in: U. Höhle, E.P. Klement (Eds.), Nonclassical Logics and Their Application to Fuzzy Subsets, Kluwer, Dordrecht, 1995, pp. 53–106. U. Höhle, A.P. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer, Dordrecht, 1999. G. Jäger, A category of L-fuzzy convergence spaces, Quaest. Math. 24 (2001) 501–517. G. Jäger, Subcategories of lattice-valued convergence spaces, Fuzzy Set. Syst. 156 (2005) 1–24. G. Jäger, Pretopological and topological lattice-valued convergence spaces, Fuzzy Set. Syst. 158 (2006) 424–435. G. Jäger, Fischer’s diagonal condition for lattice-valued convergence spaces, Quaest. Math. 31 (2008) 11–25. G. Jäger, Lattice-valued convergence spaces and regularity, Fuzzy Set. Syst. 159 (2008) 2488–2502. G. Jäger, Level spaces for lattice-valued uniform convergence spaces, Quaest. Math. 31 (2008) 255–277. G. Jäger, Lattice-valued Cauchy spaces and completion, Quaest. Math. 33 (2010) 53–74. G. Jäger, Gähler’s neighbourhood condition for lattice-valued convergence spaces, Fuzzy Set. Syst. (2012) http://dx.doi.org/10.1016/j.fss. 2012.03.014. G. Jäger, M.H. Burton, Stratified L-uniform convergence spaces, Quaest. Math. 28 (2005) 11–36. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Press, Boston, 2000. H.-J. Kowalsky, Limesräume und Komplettierung, Math. Nachrichten 12 (1954) 301–340. R. Lowen, B. Windels, AUnif: A common supercategory of pMET and Unif, Int. J. Math. Math. Sci. 21 (1998) 1–18. H. Nusser, A Generalization of probabilistic uniform spaces, Appl. Categor. Struct. 10 (2002) 81–98. D. Orpen, G. Jäger, Lattice-valued convergence spaces: extending the lattice context, Fuzzy Set. Syst. 190 (2012) 1–20. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North Holland, New York, 1983. O. Wyler, Filter space monads, regularity, completions, TOPO 1972 - General Topology and its Applications, Lecture Notes in Mathematics, vol. 378, Springer, Berlin, Heidelberg, New York, 1974, pp. 591–637.
Fuzzy Sets and Systems 210 (2013) 39 – 53 www.elsevier.com/locate/fss
Diagonal conditions for lattice-valued uniform convergence spaces Gunther Jäger Department of Statistics, Rhodes University, 6140 Grahamstown, South Africa Received 31 December 2011; received in revised form 7 May 2012; accepted 7 May 2012 Available online 17 May 2012
Abstract We define diagonal conditions for lattice-valued uniform convergence spaces and show that these conditions are preserved under initial constructions. Further, the forgetful functor from the category of stratified lattice-valued uniform convergence spaces into the category of stratified lattice-valued limit spaces maps these conditions to corresponding conditions. We especially define uniform regularity and characterize this condition by certain closures of lattice-valued filters. As an application we generalize an extension theorem for uniformly continuous mappings. © 2012 Elsevier B.V. All rights reserved. Keywords: L-fuzzy convergence; L-topology; L-filter; L-uniform convergence space; Diagonal condition; Uniform regularity; Extension theorem
1. Introduction The category of stratified lattice-valued uniform convergence spaces, SL-UCS, was introduced, for L a complete Heyting algebra, in [20]. The lattice context was extended to enriched cl-premonoids in [6,17]. The category SL-UCS is well-fibred and topological over SET and in the case that L is a complete Heyting algebra, it is Cartesian closed. SL-UCS generalizes the classical category of uniform convergence spaces UCS [4,27] in the sense that for L = {0, 1}, S{0, 1}UCS and UCS can be identified. SL-UCS contains the category of lattice-valued uniform spaces [8,9] as a reflective subcategory. Moreover, for a suitable choice of the underlying lattice, the categories of probabilistic uniform limit spaces [24] and of approach uniform spaces [23] are examples of the category of lattice-valued uniform convergence spaces [17]. In this paper, we study certain diagonal conditions in SL-UCS. In particular we define a Fischer-type diagonal condition and characterize it by means of certain neighbourhood operators. Furthermore, a concept of uniform regularity is introduced. This concept generalizes the classical concept studied by Gähler [7] and is “dual” to the Fischer-type diagonal condition. All studied conditions are finally “tied together” in an extension theorem for uniformly continuous mappings. Also this Extension Theorem generalizes a “classical” result of Gähler [7]. The paper is organised as follows. In the first section we fix the lattice background and our notation as well as the lattice-valued filter concepts that we use. In the next section we review lattice-valued uniform convergence spaces and lattice-valued limit spaces. Section 4 is then devoted to diagonal conditions à la Kowalsky and à la Fischer. Section 5 studies the concept of uniform regularity and Section 6 then presents a theorem on extension of uniformly continuous mappings. Finally we draw some conclusions. E-mail address: [email protected] 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.05.001
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
2. Preliminaries We consider in this paper enriched cl-premonoids [11,9] (L , ⱕ , ⊗, ∗). These are complete lattices (L , ⱕ ) with two algebraic operations ⊗, ∗ such that (L , ⱕ , ⊗) is a cl-premonoid, i.e. (E1) 1 ⊗ 1 ⱕ 2 ⊗ 2 whenever 1 ⱕ 2 and 1 ⱕ 2 ; (E2) ⱕ ⊗ and ⱕ ⊗ for all ∈ L; (E3) ⊗ is distributive over non-empty joins, i.e. for J ∅, j , ∈ L ( j ∈ J ): ⎛ ⎝
⎞ j⎠ ⊗ =
j∈J
⎛ ( j ⊗ ) and ⊗ ⎝
j∈J
j∈J
⎞ j⎠ =
( ⊗ j );
j∈J
and (L , ⱕ , ∗) is a GL-monoid, i.e. (E4) (L , ∗) is a commutative semigroup with as unit element; (E5) there is ∈ L such that = ∗ whenever ⱕ (divisibility [10]); (E6) ∗ is distributive over arbitrary joins, i.e. for any set J, j , ∈ L ( j ∈ J ): ⎛ ⎝
j∈J
⎞ j⎠ ∗ =
( j ∗ );
j∈J
and (E7) ∗ is dominated by ⊗, i.e. for all 1 , 2 , 1 , 2 ∈ L: (1 ⊗ 1 ) ∗ (2 ⊗ 2 ) ⱕ (1 ∗ 2 ) ⊗ (1 ∗ 2 ). From (E7) we conclude with 1 = , 2 = , 1 = , 2 = that ∗ ⱕ ⊗ for any , ∈ L. This motivates the term domination. Examples 2.1. • For a GL-monoid (L , ⱕ , ∗), the quadruple (L , ⱕ , ∗, ∗) is an enriched cl-premonoid. As a special case, with ∗ = ∧, a complete Heyting algebra, i.e. a complete lattice (L , ⱕ ) in which the frame law ( i∈J i ) ∧ = i∈J (i ∧ ) holds, is an enriched cl-premonoid, (L , ⱕ , ∧, ∧). • For a complete Heyting algebra (L , ⱕ , ∧) and a GL-monoid (L , ⱕ , ∗), the quadruple (L , ⱕ , ∧, ∗) is an enriched cl-premonoid. • Further examples are given by certain GL-monoids (L , ⱕ ,, ∗ ∗) with a monoidal mean operator ∗ [11]. • A further example is given by triangular norms on [0, 1] (see [26,21]). A t-norm is a binary operation ∗ : [0, 1] × [0, 1] −→ [0, 1] which is commutative, associative, non-decreasing in both arguments and has 1 as the unit. Important examples are the minimum t-norm, ∗ = ∧ , the product t-norm, ∗ = · and the Lukasiewicz t-norm, ∗ = ( + − 1) ∨ 0. It is clear, that for a t-norm ∗ which is left-continuous (as a function from [0, 1]2 to [0, 1]), ([0, 1], ⱕ , ∗) is a GL-monoid. All the above mentioned t-norms have monoidal mean operators such that ([0, 1], ⱕ ,, ∗ ∗) as well as ([0, 1], ⱕ , ∧, ∗) are enriched cl-premonoids. All of the enriched cl-premonoids (L , ⱕ , ⊗, ∗) mentioned above are pseudo-bisymmetric [11], i.e. for all 1 , 2 , 1 , 2 ∈ L we have (1 ∗ 1 ) ⊗ (2 ∗ 2 ) ⱕ ((1 ⊗ 2 ) ∗ (1 ⊗ 2 )) ∨ ((1 ⊗ ⊥) ∗ (1 ⊗ )) ∨ ((⊥ ⊗ 2 ) ∗ ( ⊗ 2 )). Let (L , ⱕ , ⊗, ∗) be an enriched cl-premonoid and X be a set. We denote the L-sets on X by a, b, c, . . . ∈ L X . For L-sets a, b, a j ( j ∈ J ), we extend the order relation pointwise, i.e. we write a ⱕ b if for all x ∈ X we have a(x) ⱕ b(x). Also the
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
41
operations in (L , ⱕ , ⊗, ∗) are extended pointwise by ( j∈J a j )(x) = j∈J (a j (x)), ( j∈J a j )(x) = j∈J (a j (x)), (a ⊗ b)(x) = a(x) ⊗ b(x), (a ∗ b)(x) = a(x) ∗ b(x), where x ∈ X . Further we denote for A ⊆ X and ∈ L for all x ∈ A, A (x) = ⊥ else. In particular, A is the characteristic function of A and ⊥ X is the zero function. For notions from category theory we refer to the book [1]. From here on until the end of the paper (unless stated otherwise) we will assume that (L , ⱕ ⊗, ∗) is a pseudobisymmetric enriched cl-premonoid. Let X be a set. A mapping F : L X −→ L with (F1) (F2) (F3) (Fs)
F( X ) = and F(⊥ X ) = ⊥; a ⱕ b implies F(a) ⱕ F(b); F(a) ⊗ F(b) ⱕ F(a ⊗ b); ∗ F(a) ⱕ F( X ∗ a);
(where a, b ∈ L X and ∈ L) is called a stratified L-filter on X [11]. Example. For a point x ∈ X , the point L-filter, [x], is defined by [x](a) = a(x) for a ∈ L X . Then [x] is a stratified L-filter on X. We denote the set of all stratified L-filters on X by F Ls (X ). An order relation on F Ls (X ) can be defined in the natural have F(a) ⱕ G(a). The infimum of a family of stratified L-filters {F j : j ∈ J } way: F ⱕ G if forall a ∈ L X we is then given by ( j∈J F j )(a) = j∈J (F(a)) (see [11]). For a mapping : X −→ Y and F ∈ F Ls (X ) we define ← X ← (F) ∈ F Ls (Y ) by (F)(a) = F( F Ls (Y ) we define the (a)) for←a ∈ L . Here, ←(a) = a ◦ s, see [11]. For G ∈ ← ← inverse image by (G)(a) = {G(b) : (b) ⱕ a}. Then (G) ∈ F L (X ) if and only if (b) = ⊥ X implies s G(b) = ⊥, [12]. As a special case (with the embedding i A s: A −→ X ) we obtain the trace, F A , of F ∈ F L (X ) on A ⊆ X by F A (a) = {F(b) : b| A ⱕ a}. Then F A ∈ F L (A) if and only if b| A = ⊥ A implies F(b) = ⊥, see [12]. We further denote for F ∈ F Ls (A), [F] = i A (F) ∈ F Ls (X ). Then [F](b) = F(b| A ) and [F A ] ⱖ F whenever F A ∈ F Ls (A), [12]. For a set J and G ∈ F Ls (J ) and F j ∈ F Ls (X ) for all j ∈ J we define the stratified L-diagonal filter, G(F (·) ) ∈ F Ls (X ), by G(F (·) )(a) = G(F (·) (a)) with F (·) (a)( j) = F j (a), see [15,16]. For stratified L-filters on the product space X × X , , ∈ F Ls (X × X ), we define −1 −1 X ×X ; • −1 (a) = (a a ∈ L ) with a (x, y) = a(y, x) for (x, y) ∈ X × X and ( f ) ∗ (g) with f ◦ g(x, y) = f (x, z) ∗ g(z, y) for (x, y) ∈ X × X and • ◦ (a) = X ×X
a ∈ L X ×X .
f,g∈L
: f ◦g ⱕ a
z∈X
Then −1 ∈ F Ls (X ×X ) (see [20]). Moreover, ◦ ∈ F Ls (X ×X ) if and only if f ◦g = ⊥ X ×X implies ( f )∗(g) = ⊥. (See [20] for the case of a complete Heyting algebra and [6] for a pseudo-bisymmetric enriched cl-premonoid. The condition for the “existence” of ◦ needs the assumption of pseudo-bisymmetry of the enriched cl-premonoid.) s Let L be a complete Heyting algebra and F, G ∈ F L (X ). We define the product [12], F × G, by F × G(c) = {F( f ) ∧ G(g) : f × g ⱕ c}. Here, f × g(x, y) = f (x) ∧ g(y). For the definition of the product of two stratified L-filters in more general lattice situations see [6]. We further define for F ∈ F Ls (X ) and x ∈ X the stratified L-filter Fx ∈ F Ls (X × X ) by Fx (a) = F(a(·, x)), [6]. If L is a complete Heyting algebra, then Fx = F × [x], see [20]. Lemma 2.2. Let F ∈ F Ls (X ) and denote p1 , p2 : X × X −→ X the first and second projections. Then p1 (Fx ) = F and p2 (Fx ) ⱖ [x]. Proof. We have for a ∈ L X , p1 (Fx )(a) = Fx ( p1← (a)) = F( p1← (a)(·, x)) = F(a(·)) = F(a). Further, p2 (Fx )(a) = Fx ( p2← (a)) = F( p2← (a)(·, x)) = F(a(x) X ) ⱖ a(x) = [x](a), by (Fs). Lemma 2.3. Let G ∈ F Ls (J ) and let = (1 , 2 ) : J −→ X × X . Then for j ∈ J we have (1 (G))2 ( j) = (1 × 2 )(G j ).
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
Proof. We have for b ∈ L X ×X , (1 × 2 )(G j )(b) = G j ((1 × 2 )← (b)) = G((1 × 2 )← (b)(·, j)) = G(b)(1 (·), 2 ( j)) = G(← 1 (b(·, 2 ( j)))) = 1 (G)(b(·, 2 ( j))) = (1 (G))2 ( j) (b). Lemma 2.4. Let L be a complete Heyting algebra, A ⊆ X and F ∈ F Ls (X ) such that F A ∈ F Ls (A). Then [F A × F A ] ⱖ F × F. Proof. Let b ∈ L X ×X . Then F × F(b) = {F(e) ∧ F( f ) : e × f ⱕ b}. If e × f ⱕ b, then for all x, y∈ A, e(x) ∧ f (y) ⱕ b(x, y) and hence we have e| A × f | A ⱕ b| A×A . Moreover, F A (e| A ) ⱖ F(e). Hence F × F(b) ⱕ {F A (e| A ) ∧ F A ( f | A ) : e| A × f | A ⱕ b| A×A } ⱕ F A × F A (b| A×A ) = [F A × F A ](b). We further denote the point L-filter of the point (x, y) ∈ X × X by [(x, y)] ∈ F Ls (X × X ). For a mapping : X −→ Y the product mapping × : X × X −→ Y ×Y is defined by (×)(x, y) = ((x), (y)), where (x, y) ∈ X × X . For a complete Heyting algebra, L, and F, G ∈ F Ls (X ) then we have ( × )(F × G) = (F) × (G). 3. Stratified L-uniform convergence spaces Let X be a set. A mapping : F Ls (X × X ) −→ L is called a stratified L-uniform convergence structure on X [6] ([20] for L a complete Heyting algebra) if (UC1) (UC2) (UC3) (UC4) (UC5)
∀x ∈ X : ([(x, x)]) = ; ⱕ implies () ⱕ (); () ⱕ (−1 ); () ∧ () ⱕ ( ∧ ); () ∗ () ⱕ ( ◦ ) whenever ◦ ∈ F Ls (X × X ).
The pair (X, ) is then called a stratified L-uniform convergence space. A mapping : (X, ) −→ (Y, ) between two stratified L-uniform convergence spaces is called uniformly continuous [6] if for all ∈ F Ls (X × X ) we have () ⱕ (( × )()). The category SL-UCS has as objects the stratified L-uniform convergence spaces and as morphisms the uniformly continuous mappings. It was shown in [6] that SL-UCS is well-fibred and topological over SET and that for ∗ = ⊗ = ∧ (i.e. for the case that L is a complete Heyting algebra) SL-UCS is Cartesian closed [20]. If L = {0, 1}, then stratified L-uniform convergence spaces can be identified with uniform convergence spaces [4] (with the “improved” definition due to Wyler [27]). Initial structures in SL-UCS are given as follows. Let ((X j , j )) j∈J be a family of stratified L-uniform convergence spaces and let X be a set and ( j : X −→ X j ) j∈J be a family of mappings. Then the initial SL-UCS-structure on X is given by
() = j (( j × j )()) j∈J
for ∈ F Ls (X × X ), see [6,20]. In particular, for A ⊆ X and = i A : A −→ X , we obtain the subspace (A, | A ) with | A () = ([]) for ∈ F Ls (A × A). A mapping lim : F Ls (X ) −→ L X is called a stratified L-limit structure on X [6,12,13,25] if (L1) ∀x ∈ X : lim[x]x = ; (L2) F ⱕ G implies lim F ⱕ lim G; (L3) ∀F, G ∈ F Ls (X ) : lim F ∧ lim G ⱕ lim(F ∧ G). The pair (X, lim) is then called a stratified L-limit space. A mapping : (X, lim X ) −→ (Y, limY ) between two stratified L-limit spaces is called continuous if for all F ∈ F Ls (X ) and all x ∈ X we have lim X F(x) ⱕ limY (F)((x)). The category SL-LIM with stratified L-limit spaces as objects and continuous mappings as morphisms is well-fibred and topological over SET and, in case ∗ = ⊗ = ∧, Cartesian closed [13,25]. Initial structures in SL-LIM are given as follows. Let ((X j , lim j )) j∈J be a family of stratified L-limit spaces and let X be a set and ( j : X −→ X j ) j∈J be a family of mappings. Then the initial SL-LIM-structure on X is given by
lim F(x) = lim j j (F)( j (x)) j∈J
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
43
for F ∈ F Ls (X ) and x ∈ X , see [12,13,25]. If X = X 1 × X 2 and 1 = p1 : X 1 × X 2 −→ X 1 and 2 = p2 : X 1 × X 2 − → X 2 are the projections, then we denote the initial limit structure by lim1 × lim2 . Lemma 3.1. Let (X, lim X ), (Y, limY ) ∈ |S L-L I M| and let : (X, lim X ) −→ (Y, limY ) be continuous. Then × : (X × X, lim X × lim X ) −→ (Y × Y, limY × limY ) is continuous. Proof. We denote the projections from X × X to X by p1 and p2 and the projections from Y × Y to Y by q1 and q2 . Then q1 ◦ ( × ) = ◦ p1 and q2 ◦ ( × ) = ◦ p2 . We have for ∈ F Ls (X × X ), lim X × lim X (x, y) = lim X p1 ()(x) ∧ lim X p2 ()(y) ⱕ limY ◦ p1 ()((x)) ∧ limY ◦ p2 ()((y)) = limY q1 ◦ ( × )()(q1 ◦ ( × )(x, y)) ∧ limY q2 ◦ ( × )()(q2 ◦ ( × )(x, y)) = limY × limY ( × )()((x), (y)). We can define a forgetful functor from SL-UCS to SL-LIM by ⎧ ⎪ ⎨ S L-U C S −→ S L-L I M (X, lim( )) F : (X, ) ⎪ ⎩ , where lim( )F(x) = (Fx ). This functor preserves initial structures, [20]. In particular for subspaces of (X, ) we have ( A, lim( | A )) = (A, lim( )| A ). We will later need the following result. Lemma 3.2. Let ((X j , j )) j∈J be a family of spaces in SL-UCS, let X be a set and let ( j : X −→ X j ) j∈J be a family of mappings. If (X, ) is the initial construction in SL-UCS, then for ∈ F Ls (X × X ) and (u, v) ∈ X × X we have
lim( j ) × lim( j )( j × j )()(( j × j )(u, v)). lim( ) × lim( )(u, v) ⱕ j∈J j
Proof. We denote the projections from X × X to X by p1 and p2 and the projections from X j × X j to X j by p1 and j j j p2 . Then we have p1 ◦ ( j × j ) = j ◦ p1 and p2 ◦ ( j × j ) = j ◦ p2 . Hence we have lim( ) × lim( )(u, v) = lim( ) p1 ()(u) ∧ lim( ) p2 ()(v)
= lim( j ) j ◦ p1 ()( j (u)) ∧ lim( i )i ◦ p2 ()(i (v)) j∈J
=
i∈J
j
j
lim( j ) p1 ◦ ( j × j )()( p1 ◦ ( j × j )(u, v))
j∈J
∧ ⱕ
lim( i ) p2i ◦ (i × i )()( p2i ◦ (i × i )(u, v))
i∈J j
j
j
j
[lim( j ) p1 ◦ ( j × j )()( p1 ◦ ( j × j )(u, v))
j∈J
∧ lim( j ) p2 ◦ ( j × j )()( p2 ◦ ( j × j )(u, v))]
lim( j ) × lim( j )( j × j )()(( j × j )(u, v)). = j∈J
In [14,15,25] we defined the following diagonal conditions in SL-LIM. (LK) ∀G ∈ F Ls (X ), ∀F y ∈ F Ls (X ) (y ∈ X ), ∀x ∈ X :
lim F y (y) ⱕ lim G(F (·) )(x). lim G(x) ∗ y∈X
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
(LF) ∀J, ∀ : J −→ X, ∀G ∈ F Ls (J ), ∀F j ∈ F Ls (X )( j ∈ J ), ∀x ∈ X :
lim F j (( j)) ⱕ lim G(F (·) )(x). lim (G)(x) ∗ j∈J
The condition (LK) generalizes a diagonal condition for limit spaces due to Kowalsky [22] and (LF) generalizes a diagonal condition attributed to Fischer, see [5], from the “classical case”, L = {0, 1}, to the general lattice-valued case. For more information on these conditions see [14,15,25]. A space (X, lim) ∈ |S L-L I M| is called regular [16] if it satisfies the following “dual axiom” of the axiom (LF): (LR) ∀J, ∀ : J −→ X, ∀G ∈ F Ls (J ), ∀F j ∈ F Ls (X )( j ∈ J ), ∀x ∈ X :
lim F j (( j)) ⱕ lim (G)(x). lim G(F(·) )(x) ∗ j∈J
If L is a complete Heyting algebra, then regularity can be characterized by certain closures of stratified L-filters, see [2] (and for L a complete Boolean algebra, see [16]). For F ∈ F Ls (X ) and ∈ L we define J = {(G, x) : lim G(x) ⱖ } and with this the -closure of F by F (a) = {F(b) ∧ : G(b) ∧ ⱕ a(x)∀(G, x) ∈ J } (a ∈ L X ).
Then (X, lim) is regular if and only if lim F (x) ⱖ ∧ whenever lim F(x) ⱖ . Note that in [2] the definition of F was stated with an additional condition, F (a) = {F(b) ∧ : b ∧ X ⱕ a, G(b) ∧ ⱕ a(x)∀(G, x) ∈ J }. However, if G(b) ∧ ⱕ a(x)∀(G, x) ∈ J then always b ∧ X ⱕ a, because ([x], x) ∈ J for all x ∈ X . 4. Kowalsky and Fischer type diagonal conditions in S L-U C S We first present the following adaptation of Kowalsky’s diagonal condition for a space (X, ) ∈ |S L-U C S|. We need to introduce the following notation. For ∈ F Ls (X × X ) and a family of stratified L-filters on X × X indexed by X × X , {(u,v) : (u, v) ∈ X × X } ⊆ F Ls (X × X ), we define ((·,−) ) ∈ F Ls (X × X ) by ((·,−) )(a) = ((·,−) (a)) (a ∈ L X ×X ), with (·,−) (a)(u, v) = (u,v) (a) for (u, v) ∈ X × X . (LUK) ∀ ∈ F Ls (X × X ), ∀(u,v) ∈ F Ls (X × X ), ((u, v) ∈ X × X ) :
lim( ) × lim( )(u,v) (u, v) ⱕ (((·,−) )). () ∗ (u,v)∈X ×X
Gähler [7] called a classical uniform convergence structure which satisfies (LUK) a diagonal uniform limit structure (“limesuniforme Diagonalstruktur”) . Hence we shall call (X, ) ∈ |S L-U C S| a stratified Kowalsky-diagonal L-uniform convergence space if (LUK) is satisfied. The following result seems to be new also in the classical case. Lemma 4.1. Let (X j , j ) ∈ |S L-U C S| satisfy (LUK) for all j ∈ J and let X be a set and for all j ∈ J , let j : X −→ X j be injective. Then the initial construction (X, ) satisfies (LUK). Proof. Let ∈ F Ls (X × X ) and for (u, v) ∈ X × X let (u,v) ∈ F Ls (X × X ). For j ∈ J and (x j , y j ) ∈ ( j × j )(X × X ) (x j ,y j )
we define j
= ( j × j )((u,v) ) with the unique (u, v) = ( j × j )−1 (x j , y j ). For (x j , y j ) ∈ / ( j × j )(X × X )
we define
= [(x j , y j )]. Then
(x ,y ) j j j
(x ,y )
lim( j ) × lim( j ) j j j (x j , y j ) lim( j ) × lim( j )( j × j )((u,v) )(( j × j )(u, v)) if (x j , y j ) ∈ ( j × j )(X × X ), = lim( j ) × lim( j )[(x j , y j )](x j , y j ) = otherwise.
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
45
Hence, with Lemma 3.2 and (LUK) for the spaces (X j , j ), we conclude
lim( ) × lim( )(u,v) (u, v) () ∗ (u,v)∈X ×X
ⱕ
j∈J
ⱕ
⎡
⎣ j (( j × j )()) ∗
ⱕ
⎤ lim( j ) × lim( j )( j × j )((u,v) )( j (u), j (v))⎦
(u,v)∈X ×X
⎡
⎤
⎣ j (( j × j )()) ∗
(x ,y ) lim( j ) × lim( j ) j j j (x j ,
y j )⎦
(x j ,y j )∈X j ×X j
j∈J
(·,−)
j (( j × j )()( j
)).
j∈J (x j ,y j )
Now we note that for (z, w) ∈ X × X we have, by the definition of j ( j × j )← [ j
(·,−)
(·,−)
(a)](z, w) = [ j
,
( j (z), j (w))
(a)]( j (z), j (w)) = j
(a)
= ( j × j )((z,w) )(a) = ( j × j )((·,−) )(a)(z, w). Hence we have for a ∈ L X j ×X j (·,−)
[( j × j )()]( j
)(a) = (( j × j )← [(·,−) )(a)]) = ([( j × j )((·,−) )](a))
= ((·,−) (( j × j )← (a))) = ((·,−) )(( j × j )← (a)) = ( j × j )(((·,−) ))(a), (·,−)
i.e. we have [( j × j )()]( j () ∗
) = ( j × j )(((·,−) )). Therefore
lim( ) × lim( )(u,v) (u, v) ⱕ
(u,v)∈X ×X
j (( j × j )(((·,−) ))) = (((·,−) )).
j∈J
In particular, a subspace of a stratified Kowalsky-diagonal L-uniform convergence space satisfies (LUK). Lemma 4.2. Let (X, ) ∈ |S L-U C S| satisfy the axiom (LUK). Then (X, lim( )) satisfies (LK). Proof. Let G ∈ F Ls (X ) and for each y ∈ X , let F y ∈ F Ls (X ) and let x ∈ X . Define = Gx and for (u, v) ∈ X × X define (u,v) = (F u )v . Then () = (Gx ) = lim( )G(x) and lim( ) × lim( )(u,v) (u, v) = lim( ) p1 ((u,v) )(u) ∧ lim( ) p2 ((u,v) )(v) ⱖ lim( )F u (u) ∧ lim( )[v](v) = lim( )F u (u). Moreover, we have for a ∈ L X ×X ((·,−) )(a) = Gx ((F (·) )(−) )(a) = Gx ((F (·) )(−) (a)) = G((F (·) )x (a)) = G(F (·) (a(∗, x))) = (G(F (·) ))(a(∗, x)) = (G(F (·) ))x (a). Hence we obtain lim( )G(x) ∗
lim( )F y (y) ⱕ () ∗
lim( ) × lim( )(u,v) (u, v)
(u,v)∈X ×X
y∈X
ⱕ ((
(·,−)
) = ((G(F (·) ))x ) = lim( )G(F (·) )(x),
i.e. (LK) is true. In the following we present an adaptation of Fischer’s axiom for a space (X, ) ∈ |S L-U C S|. This axiom has not been studied before in the classical case, L = {0, 1}.
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
(LUF) ∀J, ∀ : J −→ X × X, ∀ ∈ F Ls (J ), ∀ j ∈ F Ls (X × X ) ( j ∈ J ) :
lim( ) × lim( ) j (( j)) ⱕ (((·) )). (()) ∗ j∈J
We shall call a space (X, ) ∈ |S L-U C S| that satisfies (LUF) a stratified Fischer-diagonal L-uniform convergence space. Clearly, for J = X × X and = id X ×X we see that (LUF) implies (LUK). The axiom (LUF) is preserved under initial constructions. Lemma 4.3. Let (X j , j ) ∈ |S L-U C S| satisfy (LUF) for all j ∈ J and let X be a set and for all j ∈ J , let j : X −→ X j be mappings. Then the initial construction (X, ) satisfies (LUF). Proof. Let M be a set, : M −→ X × X , ∈ F Ls (M) and let for m ∈ M, m ∈ F Ls (X × X ). We define (·) j = ( j × j ) ◦ and mj = ( j × j )(m ). Then ( j ) = (( j × j )((·) )) = ( j × j )(((·) )) (see [15]). Hence we conclude
(()) ∗ lim( ) × lim( )m ((m)) ⱕ
m∈M
j (( j × j )(())) ∗
j∈J
=
j ( j ()) ∗
j∈J
ⱕ
lim( j ) × lim( j )( j × j )( )(( j × j )((m)))
m∈M
m
lim( j ) × lim( j )mj ( j (m))
m∈M (·) j (( j ))
j∈J
=
j (( j × j )(((·) ))) = (((·) )).
j∈J
In particular, products and subspaces of stratified Fischer-diagonal L-uniform convergence spaces satisfy (LUF). Because SL-UCS is a topological category and, according to Lemma 4.3, the subcategory of spaces that satisfy (LUF) is initially closed in SL-UCS, it is reflective in SL-UCS, see [1]. Lemma 4.4. Let (X, ) ∈ |S L-U C S| satisfy the axiom (LUF). Then (X, lim( )) satisfies (LF). Proof. Let J be a set, : J −→ X , G ∈ F Ls (J ) and for each j ∈ J , F j ∈ F Ls (X ) and x ∈ X . We define J = J × {x} and : J −→ X × X by ( j, x) = (( j), x). Further we define = Gx ∈ F Ls ( J) by Gx (a) = G(a(·, x)) for a ∈ L J ( j,x) j X ×X and = (F )x . We then have for a ∈ L and j ∈ J , ← (a(·, x))( j) = a(·, x)(( j)) = a(( j), x) = a(( j, x)) = ← (a)( j, x) and hence (G)x (a) = (G)(a(·, x)) = G(← (a(·, x))) = G(← (a)(·, x)) = Gx (← (a)) = (Gx )(a), i.e. we have (G)x = (). Hence lim( )(G)(x) = (((G))x ) = (()). Similarly as in the proof of Lemma 4.2 we find lim( ) × lim( ) j (( j)) ⱖ lim( )F j (( j)). Hence
lim( )F j (( j)) lim( )(G)(x) ∗ j∈J
ⱕ (()) ∗
j∈J
ⱕ (((·) )).
lim( ) × lim( ) j (( j))
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
47
Because, for a ∈ L X ×X , we have ((·) )(a) = Gx ((·) (a)) = G((·,x) (a)) = G((F (·) )x (a)) = G(F (·) (a(∗, x))) = G(F (·) )(a(∗, x)) = (G(F (·) ))x (a) we conclude (((·) )) = ((G(F (·) ))x ) = lim( )G(F (·) )(x) and (X, lim( )) satisfies (LF). We are finally going to characterize the axiom (LUF) by a neighbourhood condition. We define for ∈ L and (x, y) ∈ X × X the -level neighbourhood L-filter of (x, y) with respect to lim( ) × lim( ) by
(x,y) = . U lim( )×lim( )(x,y) ⱖ (x,y)
Clearly, U of by
(x,y)
∈ F Ls (X × X ) and U
ⱕ [(x, y)]. For ∈ F Ls (X × X ) we define the -level neighbourhood L-filter
(·)
U () = (U ). (x,y)
Then U ([(x, y)]) = U and U () ⱕ . For a similar construction for a space (X, lim) ∈ |S L-L I M| see [19]. We say that a space (X, ) ∈ |S L-U C S| satisfies the uniform neighbourhood condition if the axiom (LUG) ∀, ∈ L , ∀ ∈ F Ls (X × X ) : () ⱖ ⇒ (U ()) ⱖ ∗ is satisfied. For the classical case, L = {0, 1}, this condition goes back to the work of W. Gähler [7]. Theorem 4.5. Let (X, ) ∈ |S L-U C S|. Then (X, ) satisfies (LUF) if and only if it satisfies (LUG). Proof. Let first (X, ) satisfy (LUF) and let ∈ F Ls (X × X ) and ⱕ (). We define J = {(, (x, y)) : lim( ) × lim( )(x, y) ⱖ } and : J −→ X × X, (, (x, y)) (x, y) and for (, (x, y)) ∈ J we define (,(x,y)) = . Because lim( ) × lim( )[(x, y)](x, y) = ⱖ , the mapping is surjective. Hence K = ← () ∈ F Ls (J ) and (K) = . Furthermore,
lim( ) × lim( )(,(x,y)) ((, (x, y))) = lim( ) × lim( )(x, y) ⱖ . (,(x,y))∈J
(,(x,y))∈J
By (LUF) then ∗ ⱕ (K((·) )). We show that K((·) ) ⱕ U (). Let a ∈ L X ×X . Then K((·) )(a) = ← ()((·) (a)) = {(b) : ← (b) ⱕ (·) (a)}. If ← (b) ⱕ (·) (a), then for all (, (x, y)) ∈ J we have b(x, y) = ← (b)((, (x, y))) ⱕ ( ,(x,y)) (a) = (a). Hence
(x,y) b(x, y) ⱕ (a) = U (a) lim( )×lim( )(x,y) ⱖ
and we conclude K((·) )(a) ⱕ
(·)
(·)
{(b) : b ⱕ U (a)} ⱕ (U (a)) = U ()(a).
Hence the condition (LUG) is satisfied. Let now (X, ) satisfy (LUG). Let J be a set, : J −→ X × X , ∈ F Ls (J ) and for each j ∈ J let j ∈ F Ls (X × X ). ( j) We put = (()) and = j∈J lim( ) × lim( ) j (( j)). Then for all j ∈ J we have that U ⱕ j . By (LUG) moreover ∗ ⱕ (U (())). We show that U (()) ⱕ ((·) ). Let a ∈ L X ×X . Then (·)
(·)
U (())(a) = ()(U (a)) = (U
(a)) ⱕ ((·) (a)) = ((·) )(a).
Hence ∗ ⱕ (((·) )) and (LUF) is true. Remark 4.6. In [19] we introduced a similar condition, (LG∗ ), in the category SL-LIM: (LG∗ ) ∀, ∈ L , ∀F ∈ F Ls (X ), ∀x ∈ X : lim F(x) ⱖ ⇒ lim U (F)(x) ⱖ ∗ .
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
(·) Here, Ux = lim F (x) ⱖ F and U (F) = F(U ). We showed that this condition is equivalent to the axiom (LF). From Lemmas 4.4 and Theorem 4.5 then we conclude that if a space (X, ) ∈ |S L-U C S| satisfies (LUG), then (X, lim( )) satisfies (LG∗ ). 5. Uniform regularity A space (X, ) ∈ |S L-U C S| is called uniformly regular if it satisfies the following “dual axiom” of the axiom (LUF): ∀J, ∀ : J −→ X × X, ∀ ∈ F Ls (J ), ∀ j ∈ F Ls (X × X ) ( j ∈ J ) :
lim( ) × lim( ) j (( j)) ⱕ (()). (((·) )) ∗
(LUR)
j∈J
The axiom (LUR) generalizes the definition of uniform regularity from the classical case, L = {0, 1} as studied by Gähler [7], to the lattice-valued case. Lemma 5.1. Let (X j , j ) ∈ |S L-U C S| satisfy (LUR) for all j ∈ J and let X be a set and for all j ∈ J , let j : X −→ X j be mappings. Then the initial construction (X, ) satisfies (LUR). Proof. This proof is similar to the proof of Lemma 4.3 and is not presented. In particular, products and subspaces of uniformly regular L-uniform convergence spaces satisfy (LUR). Because SL-UCS is a topological category and, according to Lemma 5.1, the subcategory of spaces that satisfy (LUR) is initially closed in SL-UCS, it is reflective in SL-UCS, see [1]. Lemma 5.2. Let (X, ) ∈ |S L-U C S| satisfy the axiom (LUR). Then (X, lim( )) satisfies (LR). Proof. Also this proof uses methods similar to the methods of the proof of Lemma 4.4 and is not presented. We are now going to characterize uniform regularity by closures of stratified L-filters. We will restrict to the case that L is a complete Heyting algebra. The constructions and proof are simple extensions of results in [2,16]. For ∈ F Ls (X × X ) and ∈ L we define J = {(, (x, y)) : lim( ) × lim( )(x, y) ⱖ } and with this the -closure of by (a) = {(b) ∧ : (b) ∧ ⱕ a(x, y)∀(, (x, y)) ∈ J },
where a ∈ L X ×X . Then it is not difficult to show that ∈ F Ls (X × X ) and ⱕ . Moreover for ⱕ we have
ⱕ . Theorem 5.3. Let L be a complete Heyting algebra and let (X, ) ∈ |S L-U C S|. Then (X, ) is uniformly regular if and only if for all , ∈ L and all ∈ F Ls (X × X ) we have ( ) ⱖ ∧ whenever () ⱖ . Proof. Let (X, ) satisfy the axiom (LUR) and let () ⱖ . We define again J = {(, (x, y)) : ∈ F Ls (X × X ), lim( ) × lim( )(x, y) ⱖ }. For j = (, (x, y)) ∈ J we define j = and we define the mapping : J −→ X × X by ((, (x, y)) = (x, y). Then lim( ) × lim( ) j (( j)) = lim( ) × lim( )(x, y) ⱖ , i.e. ( j , (( j)) ∈ J whenever j ∈ J . We define for a ∈ L J , S(a) = (( f ) ∧ ). ( f )∧ ⱕ a( j) ∀ j=(,(x,y))∈J
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
49
Then it is not difficult to show that S ∈ F Ls (J ). Moreover, ⱕ S((·) ). To see this, let a ∈ L X ×X . Then
S((·) )(a) =
(( f ) ∧ ) =
(( f ) ∧ ) ⱖ (a) ∧ = (a).
( f )∧ ⱕ (a) ∀ j∈J
( f )∧ ⱕ (a) ∀ j∈J j
With the axiom (LUR) we conclude
((S)) ⱖ (S((·) )) ∧ lim( ) × lim( ) j (( j)) ⱖ () ∧ ⱖ ∧ . j∈J
We therefore only have to show that (S) ⱕ . Let a ∈ L X ×X . Then ← (a)((, (x, y))) = a(((, (x, y))) = a(x, y) and hence (( f ) ∧ ) (S)(a) = S(← (a)) = ( f )∧ ⱕ ← (a)((,(x,y))) ∀(,(x,y))∈J
=
(( f ) ∧ ).
( f )∧ ⱕ a(x,y) ∀(,(x,y))∈J
Hence (S)(a) ⱕ (a) and the condition of the Theorem is satisfied. Conversely, we assume that the condition of the Theorem is satisfied. Let J be a set, : J −→ X × X , ∈ F Ls (J ) and for each j ∈ J let j ∈ F Ls (X × X ). Define = j∈J lim( ) × lim( )( j)(( j)) and = (((·) )). Then
∧ ⱕ (((·) )) ). We will show that ((·) )) ⱕ (). Let a ∈ L X ×X . Then by definition
((·) ) (a) =
{((·) )( f ) ∧ : ( f ) ∧ ⱕ a(x, y) ∀(, (x, y)) ∈ J }.
We fix f ∈ L X ×X and ∈ L such that ( f ) ∧ ⱕ a(x, y) for all (, (x, y)) ∈ J . Because ( j , ( j)) ∈ J , then especially j ( f ) ∧ ⱕ a(( j)) = ← (a)( j) and it follows, by (Fs), that ((·) )( f ) ∧ ⱕ ((·) ( f ) ∧ ) ⱕ (← (a)(·)) = ()(a). Hence ∧ ⱕ (()) and the proof is complete. Corollary 5.4. Let L be a complete Heyting algebra and let (X, ) ∈ |S L-U C S|. Then (X, ) is uniformly regular if and only if for all ∈ L and all ∈ F Ls (X × X ) we have ( ) ⱖ whenever () ⱖ . If the lattice L is given by a complete Boolean algebra, then it was shown in [18] that a space (X, ) ∈ |S L-U C S| induces a stratified L-Cauchy space (X, C ) by defining, for F ∈ F Ls (X ), C (F) = (F × F). This means that (X, C ) satisfies the axioms (LC1) C ([x]) = for all x ∈ X , (LC2) F ⱕ G implies C (F) ⱕ C (G) and (LC3) C (F ∧ G) = C (F) ∧ C (G) whenever F ∨ G ∈ F Ls (X ). We will in the remainder of this section assume that L is a complete Boolean algebra. Following [3] we call (X, C ) regular if
(LCR) ∀ ∈ L ∀F ∈ F Ls (X ) : ⱕ C (F) ⇒ ⱕ C (F ). Here, F (a) = {F(b) : lim(C )G(x) ⱖ ⇒ G(b) ⱕ a(x)}, see [16], where lim(C )F(x) = C (F ∧ [x]). We note that lim(C ) = lim( ) [18]. In order to show that a uniformly regular stratified L-uniform convergence space induces a regular stratified L-Cauchy space, we introduce a new notation. We define for b ∈ L X and ∈ L the set L = {(G, x) : lim( )G(x) ⱖ } and b (x) = G(b). (G ,x)∈L
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
It is then not difficult to prove that, for a ∈ L X , we have that F (a) = {F(b) : b ⱕ a}. Similarly, for b ∈ L X ×X and ∈ L we denote J = {(, (x, y)) : lim( ) × lim( )(x, y) ⱖ } and b (x, y) = (b). (,(x,y))∈J
Then, for a ∈ L X ×X , we have that (a) = {(b) : b ⱕ a}. Note that we do not distinguish the notation for a ∈ L X and a ∈ L X ×X . The corresponding interpretation should be clear from the context.
Lemma 5.5. Let L be a complete Boolean algebra, (X, ) ∈ |S L-U C S|, ∈ L and let f 1 , f 2 ∈ L X . Then f 1 × f2 ⱕ f1 × f2 . Proof. Let (x, y) ∈ X × X . We note that (G, x), (H, y) ∈ L implies (G × H, (x, y)) ∈ J . Further, for = G × H we have that p1 () ⱖ G and p2 () ⱖ H. Hence we conclude
f 1 × f 2 (x, y) = f 1 (x) ∧ f 2 (y) G( f 1 ) ∧ = (G ,x)∈L
H( f 2 )
(H,y)∈L
ⱕ
(G( f 1 ) ∧ H( f 2 ))
(G ×H.(x,y))∈J
ⱕ
( p1 ()( f 1 ) ∧ p2 ()( f 2 ))
(,(x,y))∈J
=
(( p1← ( f 1 )) ∧ ( p2← ( f 2 )))
(,(x,y))∈J
=
( p1← ( f 1 ) ∧ p2← ( f 2 ))
(,(x,y))∈J
=
( f 1 × f 2 )
(,(x,y))∈J
= f 1 × f 2 (x, y).
Lemma 5.6. Let L be a complete Boolean algebra, (X, ) ∈ |S L-U C S|, ∈ L and let F ∈ F Ls (X ). Then F × F ⱕ F ×F . Proof. Let a ∈ L X ×X . Then, using Lemma 5.5, we obtain F × F (a) = F × F(b)
b ⱕa
=
(F( f 1 ) ∧ F( f 2 ))
b ⱕ a f1 × f2 ⱕ b
ⱕ
(F( f 1 ) ∧ F( f 2 ))
f1 × f2 ⱕ a
ⱕ
(F( f 1 ) ∧ F( f 2 )).
f1 × f2 ⱕ a
Now we note that f 1 × f 2 ⱕ a if and only if there are b, c ∈ L X such that b × c ⱕ a and f 1 ⱕ b and f 2 ⱕ c. Hence, using the frame law, we conclude
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
F × F (a) ⱕ
(F( f 1 ) ∧ F( f 2 ))
b×c ⱕ a f 1 ⱕ b, f 2 ⱕ c
=
⎛ ⎝
=
F( f 1 ) ∧
b×c ⱕ a
⎞ F( f 2 )⎠
f1 ⱕ b
51
f2 ⱕ c
(F (b) ∧ F (c))
b×c ⱕ a
= F × F (a).
Lemma 5.7. Let L be a complete Boolean algebra. If (X, ) ∈ |S L-U C S| satisfies the axiom (LUR), then (X, C ) satisfies (LCR). Proof. Let F ∈ F Ls (X ) and let ∈ L and ⱕ C (F). Then ⱕ (F × F). With Corollary 5.4 we conclude ⱕ (F × F ) and hence, by Lemma 5.6, ⱕ (F × F ) = C (F ). 6. An extension theorem for uniformly continuous mappings In this section we restrict the lattice context to complete Heyting algebras. In [16] we proved an Extension Theorem for continuous mappings. We first recall some definitions and notation. For (X, lim X ), (Y, limY ) ∈ |S L-L I M| and A ⊆ X and a mapping : A −→ Y we define, for ∈ L and x ∈ X , H A (x) = {F ∈ F Ls (X ) : F A ∈ F Ls (A) and lim X F(x) ⱖ }, FA (x) = {y ∈ Y : limY (F A )(y) ⱖ for all F ∈ H A (x)}. A subset A ⊆ X is called dense in (X, lim X ) if from x ∈ X it follows that H A (x) ∅. Further we call (X, lim X ) a T2-space if for all F ∈ F Ls (X ), x = y whenever lim X F(x) = lim X F(y) = . We will also need a stronger separation axiom. We will call (X, lim) a strong T2-space if for all F ∈ F Ls (X ), x = y whenever lim X F(x) = and lim X F(y) > ⊥. Clearly every strong T2-space is a T2-space. Theorem 6.1 (Jäger [16]). Let L be a complete Heyting algebra and let (X, lim X ) ∈ |S L-L I M| satisfy the axiom (LK) and let (Y, limY ) ∈ |S L-L I M| be a regular T2-space. Further let A ⊆ X be dense in (X, lim X ). Then a continuous mapping : (A, lim X | A ) −→ (Y, limY )has a unique continuous extension, : (X, lim X ) −→ (Y, limY ), such that | A = , if and only if for every x ∈ X , ∈L FA (x) ∅. For (X, ) ∈ |S L-U C S| we call A ⊆ X dense if A is dense in (X, lim( )). Likewise, we call (X, ) a strong T2-space, whenever (X, lim( )) is a strong T2-space. We further call (X, ) complete if for every F ∈ F Ls (X ) there is x ∈ X such that (F × F) ⱕ lim( )F(x). (See in this regard [18].) The following Theorem generalizes a result (4.12.6) of Gähler in [7], to the lattice-valued case. Theorem 6.2. Let L be a complete Heyting algebra and let (X, X ), (Y, Y ) ∈ |S L-U C S|. Further, let (X, X ) satisfy the axiom (LUG) and let (Y, Y ) be a complete, uniformly regular, strong T2-space. If A ⊆ X is dense in (X, lim( X )) and : (A, X | A ) −→ (Y, Y ) is uniformly continuous, then there is a unique uniformly continuous extension : (X, X ) −→ (Y, Y ), such that | A = . Proof. (X, lim( X )) satisfies (LF) and therefore also (LK). Further, (Y, lim( Y )) is a regular T2-space. Let x ∈ X . We show that ∈L FA (x) ∅. Because A is dense in (X, lim( X )), there is F ∈ F Ls (X ) such that F A ∈ F Ls (A) and lim( X )F(x) = . Then, using (UC3), we conclude = X (F × [x]) = ([x] × F) and, with (UC5), therefore also (see [18], Lemma 5.1) = X ((F × [x]) ◦ ([x] × F)) = X (F × F). Then also X | A (F A × F A ) = X ([F A × F A ]) ⱖ X (F × F) = . Because is uniformly continuous then Y (( × )(F A × F A )) = Y ((F A ) × (F A )) = . As (Y, Y ) is complete, there is y0 ∈ Y such that lim( Y )(F A )(y0 ) = . We show that y0 ∈ ∈L FA (x). Let G ∈ H A (x). We may assume that > ⊥. Then G A ∈ F Ls (A) and lim( X )G(x) ⱖ . Then also lim( )(F ∧ G)(x) ⱖ
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G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
and hence ((F ∧ G) × (F × G)) ⱖ . We conclude that (F ∧ G) A ∈ F Ls (A) and X | A ((F ∧ G) A × (F ∧ G) A ) ⱖ . Again, by the uniform continuity of , then Y (((F ∧ G) A ) × ((F ∧ G) A )) ⱖ . Hence, by completeness of (Y, Y ) there is y1 ∈ Y such that lim( Y )((F ∧ G) A )(y1 ) ⱖ . Therefore also lim( Y )(F A )(y1 ) ⱖ and from the strong T2 property we see that y0 = y1 . Hence y0 ∈ FA (x). We can therefore apply Theorem 6.1 and find a continuous extension : (X, lim( X )) −→ (Y, lim( Y )) with | A = . We show that : (X, X ) −→ (Y, Y ) is uniformly continuous. Let ∈ F Ls (X × X ) and let X () = . By (LUG) then also ⱕ X (U ()). Then U () A×A ∈ F Ls (A × A). To see this, let b| A×A = ⊥ A×A . Then, for (x, y) ∈ X × X ,
(x,y) U (b) ⱕ (b) = ⊥, lim( X )×lim( X )(x,y) ⱖ ,( A×A )=
because from the denseness of A we find F, G ∈ F Ls (X ) such that lim( X )[F A ](x) ⱖ and lim( X )[G A ](y) ⱖ . It is not difficult to see that = [F A ] × [G A ] then satisfies lim( X ) × lim( X )(x, y) ⱖ and ( A×A ) = . Hence (·,−) U ()(b) = (U (b)) = (⊥ X ×X ) = ⊥. Then X | A (U () A×A ) = X ([U () A×A ]) ⱖ X (U ()) ⱖ . From the uniform continuity of and because (Y, Y ) is uniformly regular we conclude
ⱕ Y (( × )(U () A×A )) ⱕ Y (( × )(U () A×A ) ).
We finally show that ( × )(U () A×A ) ⱕ ( × )(). Let b ∈ L Y ×Y . Then, with JY = {(, (y1 , y2 )) : lim( Y ) × lim( Y )(y1 , y2 ) ⱖ },
( × )(U () A×A ) (b) (·,−) = {(U ([( × )← (c)]∗ )) ∧ : (c) ∧ ⱕ b(y1 , y2 )∀(, (y1 , y2 )) ∈ JY }, where we denote ←
∗
[( × ) (c)] (x1 , x2 ) =
( × )← (c)(x1 , x2 ) if (x1 , x2 ) ∈ A × A,
otherwise.
It is easy to show that, with a similar notation, [( × )← (c)| A×A ]∗ = [( × )← (c)]∗ . We fix now c ∈ L Y ×Y and ∈ L such that (c) ∧ ⱕ b(y1 , y2 ) for all (, (y1 , y2 )) ∈ JY . Then (x1 ,x2 )
U
=
([( × )← (c)]∗ )
([( × )← (c)| A×A ]∗ )
lim( X )×lim( X )(x1 ,x2 ) ⱖ
ⱕ
[ ]([( × )← (c)| A×A ]∗ )
lim( X )×lim( X )[ ](x1 ,x2 ) ⱖ , ∈F Ls (A×A)
=
(( × )← (c)| A×A )
lim( X )×lim( X )[ ](x1 ,x2 ) ⱖ , ∈F Ls (A×A)
=
[ ](( × )← (c))
lim( X )×lim( X )[ ](x1 ,x2 ) ⱖ , ∈F Ls (A×A)
=
( × )([ ])(c).
lim( X )×lim( X )[ ](x1 ,x2 ) ⱖ , ∈F Ls (A×A)
If lim( X ) × lim( X )[ ](x1 , x2 ) ⱖ , then by continuity of × with respect to lim( X ) × lim( X ) and lim( Y ) × lim( Y ) we have (( × )([ ]), ((x1 ), (x2 ))) ∈ JY . By choice of c and then ( × )([ ])(c) ∧ ⱕ ( × (x ,x ) )← (b)(x1 , x2 ). Hence U 1 2 ([( × )← (c)]∗ ) ∧ ⱕ ( × )← (b)(x1 , x2 ) and we obtain with the stratification condition (Fs)
(·,−)
( f × )(U () A×A ) (b) ⱕ (U This completes the proof.
([( × )← (c)]∗ ∧ ) ⱕ (( × )← (b)) = ( × )()(b).
G. Jäger / Fuzzy Sets and Systems 210 (2013) 39 – 53
53
7. Conclusions We defined and studied several diagonal conditions for lattice-valued uniform convergence spaces. We especially showed that a Fischer-type diagonal condition is equivalent to a neighbourhood condition which was studied for L = {0, 1} by Gähler [7]. Uniform regularity was defined and characterized by closures of stratified L-filters. All the studied notions were finally used to obtain an extension theorem for uniformly continuous mappings. In the classical case, it was shown in [7] that a uniform space satisfies all these diagonal conditions, but that, however, the diagonal conditions do not characterize the uniform spaces among the uniform convergence spaces. This is in contrast to the situation in the category LIM of limit spaces, where e.g. Fischer’s diagonal condition characterizes topological limit spaces among the limit spaces. It is at present not known if lattice-valued uniform spaces [8,9] are uniformly regular or if they satisfy the axiom (LUF). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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