Continuous [0,1]-lattices and injective [0,1]-approach spaces
aa r X i v : . [ m a t h . GN ] F e b Continuous [0 , , Junche Yu, Dexue Zhang
School of Mathematics, Sichuan University, Chengdu 610064, [email protected], [email protected]
Abstract
In 1972, Dana Scott proved a fundamental result on the connection between orderand topology which says that injective T spaces are precisely continuous lattices en-dowed with Scott topology. This paper investigates whether this is true in the enrichedcontext, where the enrichment is a quantale obtained by equipping the interval [0 ,
1] witha continuous t-norm. It is shown that for each continuous t-norm, the specialization [0 , , X is a continuous [0 , , X coincides with the Scott [0 , , Keywords
Continuous t-norm, [0 , , , , , MSC(2020)
Given a topological space X , the specialization order of X refers to the order relation ≤ defined by x ≤ y if x is in the closure of { y } [13]. It is obvious that X is a T space if andonly if the specialization order satisfies the axiom of anti-symmetry. Taking specializationdefines a functor Ω : Top −→ Ord from the category of topological spaces to that of ordered sets and order-preserving maps.The specialization order of any space that satisfies the separation axiom of T is always thediscrete order, so, for a general space, the functor Ω forgets much information of that space.There are spaces for which Ω “forgets nothing”. If X is an Alexandroff space, that means,every point of X has a smallest neighborhood, then the structure of X can be recovered fromits specialization order. Actually, the functor Ω has a left adjointΓ : Ord −→ Top that sends each ordered set ( X, ≤ ) to the space having all the lower sets of ( X, ≤ ) as closedsets. Alexandroff spaces are then those spaces X for which ΓΩ( X ) = X . These spaces arenot very interesting, however.A celebrated result of Scott [42] says that the structure of every injective T space isencoded in its specialization order. In the words of Scott, these spaces “are at the same time1omplete lattices whose topology is determined by the lattice structure in a special way”.That special way means the functor Σ : Ord ↑ −→ Top from the category of ordered sets and Scott continuous maps (i.e., maps that preserve directedjoins) to that of topological spaces, topologizing each ordered set with its Scott topology.Precisely, a T space X is injective if, and only if, the specialization order of X is a continuouslattice and the topology of X coincides with the Scott topology of its specialization order.The functor Σ, which throws new light on the connections between order and topology [42],plays a prominent role in domain theory [13, 14].In this paper, we address the question: Is there an enriched version of the result of Scott?That means, we study the relationship between enriched orders and enriched topologies, inparticular, enriched version of Scott topology.Enriched ordered sets Enriched topological spaces / / Enriched ordered sets Enriched topological spaces o o We take • quantale-valued orders for enriched orders, • quantale-valued approach spaces for enriched topological spaces, and • forward Cauchy weights for enriched directed lower sets.We haste to remark that there are other choices for enriched topological spaces andenriched directed lower sets.It is proved that for the quantale obtained by endowing the interval [0 ,
1] with a continuoust-norm, the specialization [0 , , X isa continuous [0 , X is completely determined by its specialization[0 , , , raison d’ˆetre of the paper.We end the introduction with a few words on related works.In [18, 19] Hofmann presents a new approach to the characterization of injective approachspaces. We note that approach spaces are essentially [0 , cocomplete spaces . For a comprehensiveaccount of story of topological spaces being viewed as categories or lax algebras, we referto the monograph [20]. Further, such spaces are characterized in [15] as continuous latticesequipped with a [0 , ∞ ] -action that satisfies certain conditions. The main difference betweenthese works and this paper is that here the focus is on an enriched version of Scott topology .So, this paper is a continuation and a strengthening of [33]. All continuous t-norms are takeninto consideration here, whereas [33] deals with only the product t-norm.2n [50], a frame-valued version the result of Scott is established, where, fuzzy topologicalspaces (valued in a frame) play the role of enriched topological spaces and flat ideals (see[30, Remark 3.5] for more information) play the role of directed lower sets. Compared with[50], we have a different kind of quantales (in which the semigroup operation need not beidempotent), a different kind of enriched topological spaces, and a different kind of enricheddirected lower sets.
In this section we recall some basic notions needed in this paper and fix some notations.By a quantale we mean a commutative and unital one in the sense of [41]. Explicitly, aquantale Q = ( Q , & , k )is a commutative monoid with k being the unit, such that the underlying set Q is a completelattice with a bottom element 0 and a top element 1, and that the multiplication & distributesover arbitrary joins. We say that Q is integral if the unit k coincides with the top element1. The multiplication & determines a binary operator → , sometimes called the implicationoperator of &, via the adjoint property: p & q ≤ r ⇐⇒ q ≤ p → r. In the language of category theory, a quantale is precisely a small, complete and symmetricmonoidal closed category [6, 24, 31].Quantales obtained by endowing the unit interval [0 ,
1] with a continuous t-norm are ofparticular interest in this paper. A continuous t-norm [25] is, actually, a continuous map& : [0 , −→ [0 ,
1] that makes ([0 , , & ,
1) into a quantale. Such quantales play a decisiverole in the BL-logic of H´ajek [16].Basic continuous t-norms and their implication operators are listed below:(1) The G¨odel t-norm: x & y = min { x, y } ; x → y = ( , x ≤ y,y, x > y. The implication operator → of the G¨odel t-norm is continuous except at ( x, x ), x < x & P y = xy ; x → y = ( , x ≤ y,y/x, x > y. The implication operator → of the product t-norm is continuous except at (0 , , , & P ,
1) is isomorphic to Lawvere’s quantale ([0 , ∞ ] op , + ,
0) [31].(3) The Lukasiewicz t-norm: x & L y = max { , x + y − } ; x → y = min { − x + y, } . The implication operator → of the Lukasiewicz t-norm is continuous on [0 , .3et & be a continuous t-norm. An element p ∈ [0 ,
1] is idempotent if p & p = p . Proposition 2.1. ([25, Proposition 2.3])
Let & be a continuous t-norm on [0 , and p be anidempotent element of & . Then x & y = min { x, y } whenever x ≤ p ≤ y . It follows immediately that y → x = x whenever x < p ≤ y for some idempotent p .Another consequence of Proposition 2.1 is that for any idempotent elements p, q with p < q ,the restriction of & to [ p, q ], which is also denoted by &, makes [ p, q ] into a commutativequantale with q being the unit element. The following theorem, known as the ordinal sumdecomposition theorem , is of fundamental importance. Theorem 2.2. ([25, 40])
Let & be a continuous t-norm. If a ∈ [0 , is non-idempotent,then there exist idempotent elements a − , a + ∈ [0 , such that a − < a < a + and the quantale ([ a − , a + ] , & , a + ) is either isomorphic to ([0 , , & L , or to ([0 , , & P , . Conversely, foreach set of disjoint open intervals { ( a n , b n ) } n of [0 , , the binary operator x & y := a n + ( b n − a n ) & n (cid:16) x − a n b n − a n , y − a n b n − a n (cid:17) , ( x, y ) ∈ [ a n , b n ] , min { x, y } , otherwise is a continuous t-norm, where each & n is a continuous t-norm on [0 , . Proposition 2.3. ([29])
For a continuous t-norm & on [0 , , the following statements areequivalent: (1) The implication → : [0 , × [0 , −→ [0 , is continuous at each point off the diagonal { ( x, x ) | x ∈ [0 , } . (2) For each non-idempotent element a ∈ [0 , , the quantale ([ a − , a + ] , & , a + ) is isomorphicto ([0 , , & P , whenever a − > . We say that a continuous t-norm & on [0 ,
1] satisfies the condition (S) if it satisfies theequivalent conditions in Proposition 2.3.A Q -valued order , a Q -order for short, is a map α : X × X −→ Q such that for all x, y, z ∈ X , α ( x, x ) ≥ k and α ( y, z ) & α ( x, y ) ≤ α ( x, z ) . The pair (
X, α ) is call a Q -ordered set (or, a Q -category in the language of enriched categories).It is customary to write X for the pair ( X, α ) and write X ( x, y ) for α ( x, y ).If α is a Q -order on X , it follows from the commutativity of & that α op ( x, y ) := α ( y, x )is also a Q -order on X , called the opposite of α .We say that a map f : X −→ Y between Q -ordered sets preserves Q -order if X ( x, y ) ≤ Y ( f ( x ) , f ( y )) for all x, y ∈ X . A Q -order-preserving map is fully faithful if X ( x, y ) = Y ( f ( x ) , f ( y )) for all x, y ∈ X . Q -ordered sets and Q -order-preserving maps constitute acategory Q - Ord . Remark 2.4.
In 1973, Lawvere [31] argued that each arbitrary closed category, a quantalein particular, can be thought of as the table of truth-values of a generalized logic, and thetheory of enriched categories is of a logic nature. This idea has led to the study of quantale-enriched categories, generalized metric spaces in particular, as quantitative domains . A fewof the early works are listed here: [2, 5, 9, 10, 46].4 xample 2.5.
For all x, y ∈ Q , let α L ( x, y ) = x → y. Then α L is a separated Q -order on Q . We denote the opposite of α L by α R ; that is, α R ( x, y ) = y → x. Both ( Q , α L ) and ( Q , α R ) play important roles in this paper. Example 2.6.
For each set X , the mapsub X : Q X × Q X −→ Q , sub X ( φ, ψ ) = ^ x ∈ X φ ( x ) → ψ ( x )is a Q -order on the set Q X , known as the fuzzy inclusion Q -order. If X is a singleton set,then ( X, sub X ) degenerates to the Q -ordered set ( Q , α L ).The underlying order of a Q -ordered set X refers to the order ≤ given by x ≤ y if k ≤ X ( x, y ) . For convenience, for each Q -ordered set X , we shall write X for the ordered set obtained byequipping X with the underlying order.Two elements x, y of a Q -ordered set X are isomorphic if k ≤ X ( x, y ) ∧ X ( y, x ) . A Q -ordered set X is separated if isomorphic elements are identical; that is, k ≤ X ( x, y ) ∧ X ( y, x ) ⇒ x = y. Said differently, X is separated if X satisfies the axiom of anti-symmetry.Taking underlying order defines a functor from Q - Ord to the category of ordered sets andorder-preserving maps ι : Q - Ord −→ Ord . The functor ι has a left adjoint ω : Ord −→ Q - Ord that sends an order ≤ on a set X to the Q -order α on X given by α ( x, y ) = k if x ≤ y , and α ( x, y ) = 0 if x y .The category Q - Ord is complete and cocomplete; for example, the product Q i X i of a setof Q -ordered sets { X i } i is given by the Cartesian product of the sets { X i } i and Y i X i ( ~x, ~y ) = ^ i X i ( x i , y i )for all ~x, ~y ∈ Q i X i .Let X, Y be Q -ordered sets; let f : X −→ Y and g : Y −→ X be maps. We say that f isleft adjoint to g (and/or, g is right adjoint to f ) and write f ⊣ g , if Y ( f ( x ) , y ) = X ( x, g ( y ))for all x ∈ X and y ∈ Y . This is a special case of enriched adjunctions in enriched categorytheory [6, 24]. 5 heorem 2.7. ([44, page 295]) Let f : X −→ Y and g : Y −→ X be a pair of maps between Q -ordered sets. Then f is left adjoint to g if and only if the following conditions are satisfied: (1) Both f and g preserve Q -order. (2) The map f : X −→ Y is left adjoint to g : Y −→ X ; that is, for all x ∈ X and y ∈ Y , f ( x ) ≤ y ⇐⇒ x ≤ g ( y ) . Suppose that X is a Q -ordered set. A weight of X , also known as a lower fuzzy set of X ,is a map φ : X −→ Q such that for all x, y ∈ X , φ ( y ) & X ( x, y ) ≤ φ ( x ) . Said differently, a weight of X is a Q -order-preserving map φ : X −→ ( Q , α R ). The weightsof X constitute a Q -ordered set P X with P X ( φ , φ ) := sub X ( φ , φ ) . For each x ∈ X , X ( − , x ) is a weight of X (the principal lower fuzzy set generated by x )and we have the following: Lemma 2.8 (Yoneda lemma) . Let X be a Q -ordered set and φ be a weight of X , then P X ( X ( − , x ) , φ ) = φ ( x ) . The above lemma is a special case of the Yoneda lemma in enriched category theory, seee.g. [6, 43]. The Yoneda lemma entails that the map y : X −→ P X, x X ( − , x )is fully faithful, hence an embedding if X is separated. By abuse of language, we call it the Yoneda embedding no matter X is separated or not.Each Q -order-preserving map f : X −→ Y gives rise to an adjunction between P X and P Y . Precisely, the map f → : P X −→ P Y, f → ( φ )( y ) = _ x ∈ X φ ( x ) & Y ( y, f ( x ))is left adjoint to f ← : P Y −→ P X, f ← ( ψ )( x ) = ψ ( f ( x )) . Let X be a Q -ordered set; let a be an element and φ be a weight of X . We say that a isa supremum of φ if X ( a, y ) = P X ( φ, X ( − , y ))for all y ∈ X . Suprema of a weight, if exist, are unique up to isomorphism. We say that X is cocomplete if every weight of X has a supremum. It is clear that X is cocomplete if andonly if the Yoneda embedding y : X −→ P X has a left adjoint. Example 2.9. (Examples 2.11 and 2.12 in [30]) Let X be Q -ordered set.(1) If ψ : X −→ ( Q , α L ) is a Q -order-preserving map, then for each weight λ of X ,sup ψ → ( λ ) = _ x ∈ X λ ( x ) & ψ ( x ) . (2.i)62) If φ is a weight of X , viewed as a Q -order-preserving map X −→ ( Q , α R ), then for eachweight λ of X , sup φ → ( λ ) = sub X ( λ, φ ) . (2.ii)We say that a Q -ordered set X is complete if X op is cocomplete. It is known, see e.g. [43],that a Q -ordered set is cocomplete if and only if it is complete and that a Q -order-preservingmap f : X −→ Y between cocomplete Q -ordered sets is a left adjoint if and only if f preservessuprema in the sense that f (sup φ ) = sup f → ( φ ) for each weight φ of X .A Q -ordered set is called a complete Q -lattice if it is separated and complete (or equiva-lently, cocomplete). The category of complete Q -lattices and left adjoints is denoted by Q - Sup . For each Q -ordered set X , P X is a complete Q -lattice. The left adjoint of the Yonedaembedding y P X : P X −→ PP X is given by m X : PP X −→ P X, Φ _ φ ∈P X Φ( φ ) & φ. The correspondence f : X −→ Y f → : P X −→ P Y defines a functor P : Q - Ord −→ Q - Sup that is left adjoint to the forgetful functor U : Q - Sup −→ Q - Ord [43]. We denote by P = ( P , m , y )the monad on Q - Ord arising from the adjunction
P ⊣ U . Explicitly, for a Q -ordered set X , • P X is the complete Q -lattice of all weights of X ; • the unit is the Yoneda embedding y X : X −→ P X ; • the multiplication m X : PP X −→ P X is the left adjoint of the Yoneda embedding y P X : P X −→ PP X .The category of P -algebras and P -homomorphisms is precisely the category of Q -completelattices and left adjoints [43]; that is, Q - Sup = P - Alg . In particular, the forgetful functor Q - Sup −→ Q - Ord is monadic. The following conclusion isan instance of [29, Proposition 3.1], which is quite likely to have appeared somewhere else.
Proposition 2.10.
Every retract of a complete Q -lattice in Q - Ord is a complete Q -lattice. While a Q -ordered set is a Q -enriched ordered set, a Q -approach space is a Q -enriched topo-logical space. 7 efinition 3.1. Let Q = ( Q , & , k ) be a quantale. A Q -valued approach structure on a set X is a map δ : X × X −→ Q subject to the following conditions: for all x ∈ X , and A, B ∈ X ,(A1) δ ( x, { x } ) ≥ k ;(A2) δ ( x, ∅ ) = 0;(A3) δ ( x, A ∪ B ) = δ ( x, A ) ∨ δ ( x, B );(A4) δ ( x, A ) ≥ δ ( x, B ) & V b ∈ B δ ( b, A ).The pair ( X, δ ) is called a Q -valued approach space, a Q -approach space for simplicity. Remark 3.2.
In a Q -approach space, the value δ ( x, A ) measures how close a point is toa subset. When Q is the boolean algebra ( { , } , ∧ , Q -approach space is precisely atopological space with δ ( x, A ) denoting whether x is in the closure of A . So, Q -approachspaces can be thought of as “quantitative topological spaces”; actually, they are called Q -valued topological spaces in [26, 27]. Q -approach spaces have appeared in the literature underdifferent names. In [12, 23], Q -approach structures are called fuzzy contiguity relations . In thecase that Q is Lawvere’s quantale ([0 , ∞ ] op , + , approach spaces . If Q is the unit interval equipped with a left continuoust-norm, such spaces are probabilistic topological spaces in the sense of Frank [11] with someextra requirements.A map f : ( X, δ X ) −→ ( Y, δ Y ) between Q -approach spaces is continuous if for all x ∈ X and A ⊆ X , δ X ( x, A ) ≤ δ Y ( f ( x ) , f ( A )) . Q -approach spaces and continuous maps constitute a category Q - App , an enriched version of the category of topological spaces.It is shown in [27] that if the underlying lattice of Q has enough coprimes, then Q - App is topological over the category of sets in the sense of [1], in particular, it is complete andcocomplete.
Example 3.3.
Let Q be a linearly ordered quantale. Then the map δ K : Q × Q −→ Q , δ K ( x, A ) = ( inf A → x, A = ∅ , , A = ∅ is a Q -approach structure on Q . The space K := ( Q , δ K ) plays an important role in this paper.This space was introduced by Lowen, see e.g. [36, Examples 1.8.33 (2)], when Q is Lawvere’squantale ([0 , ∞ ] op , + , Proposition 3.4.
The space K is an initially dense object in the category Q - App .Proof.
For each Q -approach space ( X, δ ) and each subset A of X , it follows from (A4) that δ ( − , A ) : ( X, δ ) −→ ( Q , δ K ) is a continuous map. Then, it is readily verified that { ( X, δ ) δ ( − ,A ) / / ( Q , δ K ) } A ⊆ X is an initial source. 8s one might expect, the relation between Q -ordered sets and Q -approach spaces is anal-ogous to that between ordered sets and topological spaces. For each Q -approach space ( X, δ ),the Q -relation Ω( δ ) : X × X −→ Q , Ω( δ )( x, y ) = δ ( x, { y } )is a Q -order on X , called the specialization Q -order of ( X, δ ). Assigning to each Q -approachspace its specialization Q -order gives rise to a functorΩ : Q - App −→ Q - Ord . This functor has a left adjoint Γ : Q - Ord −→ Q - App that sends each Q -ordered set ( X, α ) to the Q -approach space ( X, Γ( α )) given byΓ( α )( x, A ) = _ a ∈ A α ( x, a ) . It is clear that Γ is a full and faithful functor.The adjunction Γ ⊣ Ω is an analogue of that between the categories of ordered sets andtopological spaces mentioned in the introduction. In the classic case, the left adjoint sendsevery ordered set to its Alexandroff topology and the right adjoint sends each topologicalspace to its specialization order.Now we postulate a class of Q -approach spaces in terms of the specialization Q -order.These spaces are an extension of T topological spaces to the quantale-enriched context. Definition 3.5. A Q -approach space ( X, δ ) is called separated if the specialization Q -orderΩ( δ ) is separated.Given a Q -approach space ( X, δ ) and a subset Y of X , it is clear that restricting thedomain of δ to Y × Y yields a Q -approach structure on Y , the resulting space is called asubspace of ( X, δ ).A Q -approach space ( Z, δ Z ) is said to be injective if for each Q -approach space ( X, δ X )and each continuous map f from a subspace ( Y, δ Y ) of ( X, δ X ) to ( Z, δ Z ), there is a continuousmap f : ( X, δ X ) −→ ( Z, δ Z ) that extends f .Since Q -approach spaces are topological spaces enriched over Q , the main question of thispaper is: Question 3.6.
Are the separated and injective Q -approach spaces precisely continuous lat-tices enriched over Q as in the classical situation?We’ll show that in the case that Q is the interval [0 ,
1] together with a continuous t-norm,the answer depends on the structure of the continuous t-norm which plays the role of theconnective conjunction in fuzzy logic. Q -cotopological spaces In this section we show that if the quantale Q is the interval [0 ,
1] equipped with a continuoust-norm, then a separated Q -approach space is injective if and only if it is a retract of somepower of the space K in Example 3.3. We do this by showing that for such a quantale, thecategory of Q -approach spaces is isomorphic to the category of strong Q -cotopological spaces.9 notation first. Let X be a set and let A be a subset of X . For each p ∈ Q , we write p A for the map X −→ Q given by p A ( x ) = ( p, x ∈ A, , x / ∈ A. By a quasi- Q -cotopology on a set X we mean a subset τ of Q X subject to the followingconditions:(C1) 0 X ∈ τ ;(C2) φ ∨ ψ ∈ τ for all φ, ψ ∈ τ ;(C3) V i φ i ∈ τ for each subset { φ i } of τ .The pair ( X, τ ) is called a quasi- Q -cotopological space; elements of τ are called closed sets.Let ( X, τ ) be a quasi- Q -cotopological space. The closure operator of ( X, τ ) is the map( − ) − : Q X −→ Q X given by µ = ^ { φ ∈ τ : µ ≤ φ } . The closure operator satisfies the following conditions: for all λ, µ ∈ Q X ,(cl1) 0 X = 0 X ;(cl2) µ ≥ µ ;(cl3) λ ∨ µ = λ ∨ µ ;(cl4) µ = µ .It is clear that quasi- Q -cotopologies on X correspond bijectively to operators Q X −→ Q X satisfying (cl1)-(cl4). Definition 4.1. A Q -cotopology on a set X is a quasi- Q -cotopology τ that is stratified inthe sense that(C4) p → φ ∈ τ for all p ∈ Q and φ ∈ τ .Condition (2) in the following proposition explains why the term Q -cotopology is reservedfor quasi- Q -cotopology satisfying (C4): the closure operator of a Q -cotopological space shouldpreserve Q -order. Proposition 4.2. ([7])
Let ( X, τ ) be a quasi- Q -cotopological space. Then the following state-ments are equivalent: (1) τ is stratified, hence ( X, τ ) is a Q -cotopological space. (2) The closure operator preserves Q -order, i.e., sub X ( λ, µ ) ≤ sub X ( λ, µ ) for all λ, µ ∈ Q X . (3) The closure operator satisfies (cl5) p & µ ≤ p & µ for all p ∈ Q and µ ∈ Q X .
10 map f : ( X, τ X ) −→ ( Y, τ Y ) between Q -cotopological spaces is continuous if λ ◦ f ∈ τ X for all λ ∈ τ Y . Q -cotopological spaces and continuous maps constitute a topological category Q - CTop . Given a Q -cotopological space ( X, τ ), the map ζ ( τ ) : X × X −→ Q , ζ ( τ )( x, A ) = k A ( x )is easily verified to be a Q -approach structure on X . Assigning to each Q -cotopological space( X, τ ) the Q -approach space ( X, ζ ( τ )) gives rise to a functor ζ : Q - CTop −→ Q - App . The specialization Q -order Ω( τ ) of a Q -cotopological space ( X, τ ) is defined to be thespecialization Q -order of the Q -approach space ζ ( X, τ ); that is, Ω( τ )( x, y ) = k y ( x ). We saythat ( X, τ ) is a T space if Ω( τ ) is separated, or equivalently, if ζ ( X, τ ) is separated.A Q -cotopology on a set X is essentially a map τ : X × Q X −→ Q subject to the following conditions:(1) τ ( x, λ ) ≥ λ ( x );(2) τ ( x, X ) = 0;(3) τ ( x, λ ∨ µ ) = τ ( x, λ ) ∨ τ ( x, µ );(4) τ ( x, λ ) ≥ τ ( x, µ ) & sub X ( µ, τ ( − , λ )).The Q -approach space ζ ( X, τ ) is obtained by restricting the domain of τ : X × Q X −→ Q to X × X . Thus, the functors Q - CTop ζ / / Q - App Ω / / Q - Ord are obtained by composing the corresponding structure maps with the embeddings X × X ( x,y ) ( x, { y } ) / / X × X ( x,A ) ( x,k A ) / / X × Q X , respectively. Conversely, to make a Q -approach space into a Q -cotopological space, we needto extend the domain of δ : X × X −→ Q to X × Q X . It seems hard to do so for a generalquantale, however, this is possible when Q is linearly ordered. Lemma 4.3.
Let Q be a linearly ordered quantale. Then for each Q -approach space ( X, δ ) ,the set κ ( δ ) of all continuous maps ( X, δ ) −→ K is a Q -cotopology on X .Proof. We leave it to the reader to check that κ ( δ ) satisfies the conditions (C1) and (C3).Here we check that it satisfies the conditions (C2) and (C4).Let λ : ( X, δ ) −→ K be a continuous map and let p ∈ Q . Then for each x ∈ X and each A ⊆ X , if A is empty, it holds trivially that δ ( x, A ) ≤ δ K (( p → λ )( x ) , ( p → λ )( A ));11f A = ∅ , then δ ( x, A ) ≤ inf λ ( A ) → λ ( x ) ≤ ( p → inf λ ( A )) → ( p → λ ( x ))= δ K (( p → λ )( x ) , ( p → λ )( A )) . Thus, p → λ : ( X, δ ) −→ K is a continuous map and κ ( δ ) satisfies (C4).To show that κ ( δ ) satisfies (C2), we make use of the assumption that Q is linearly ordered.Assume that λ and µ are continuous maps from ( X, δ ) to ( Q , δ K ). Let x ∈ X and A ⊆ X .We need to show that δ ( x, A ) ≤ δ K ( λ ( x ) ∨ µ ( x ) , ( λ ∨ µ )( A )) . Let B = { a ∈ A | µ ( a ) ≤ λ ( a ) } ; C = { a ∈ A | λ ( a ) ≤ µ ( a ) } . Since Q is linearly ordered, it follows that A = B ∪ C , thus, δ ( x, A ) = δ ( x, B ) ∨ δ ( x, C ) ≤ δ K ( λ ( x ) , λ ( B )) ∨ δ K ( µ ( x ) , µ ( C )) ≤ δ K ( λ ( x ) ∨ µ ( x ) , λ ( B )) ∨ δ K ( λ ( x ) ∨ µ ( x ) , µ ( C ))= δ K ( λ ( x ) ∨ µ ( x ) , ( λ ∨ µ )( A )) , which completes the proof. Example 4.4.
Let Q be a linearly ordered quantale; let ( X, α ) be a Q -ordered set. Since foreach map λ : X −→ Q , we have λ : ( X, Γ( α )) −→ K is continuous ⇐⇒ ∀ x ∈ X, ∀ A ⊆ X, Γ( α )( x, A ) ≤ δ K ( λ ( x ) , λ ( A )) ⇐⇒ ∀ x ∈ X, ∀ A = ∅ , W a ∈ A X ( x, a ) ≤ inf λ ( A ) → λ ( x ) ⇐⇒ ∀ x, a ∈ X, X ( x, a ) ≤ λ ( a ) → λ ( x ) ⇐⇒ λ : X −→ ( Q , α R ) preserves Q -order , it follows that the closed sets of κ ◦ Γ( α ) are exactly the weights of ( X, α ). Example 4.5.
Let Q = ([0 , , & ,
1) with & being a left continuous t-norm. Then a map λ : [0 , −→ [0 ,
1] is a closed set of the Q -cotopological space ([0 , , κ ( δ K )) if and only if • y → x ≤ λ ( y ) → λ ( x ) for all x, y ∈ [0 , • λ is right continuous.By definition, we have λ ∈ κ ( δ K ) ⇐⇒ λ : ([0 , , δ K ) −→ ([0 , , δ K ) is continuous ⇐⇒ ∀ x ∈ X, ∀ A = ∅ , δ K ( x, A ) ≤ δ K ( λ ( x ) , λ ( A )) ⇐⇒ ∀ x ∈ X, ∀ A = ∅ , inf A → x ≤ inf λ ( A ) → λ ( x ) . Suppose that λ ∈ κ ( δ K ). Letting A = { y } gives that y → x ≤ λ ( y ) → λ ( x ), in particular, λ is non-decreasing. Letting x = inf A for a nonempty subset gives that λ (inf A ) = inf λ ( A ),hence λ is right continuous. The converse implication is clear.12 roposition 4.6. Let Q be a linearly ordered quantale. Then, the assignment ( X, δ ) ( X, κ ( δ )) gives rise to a functor κ : Q - App −→ Q - CTop that is left adjoint and right inverse to ζ : Q - CTop −→ Q - App . Proof.
First, we prove a useful fact: for each Q -approach space ( X, δ ) and each subset A of X , the map δ ( − , A ) : X −→ Q , x δ ( x, A )is a closed set of ( X, κ ( δ )) and it is the closure of k A in ( X, κ ( δ )); that is, k A = δ ( − , A ).On the one hand, it follows from (A4) that δ ( − , A ) : ( X, δ ) −→ ( Q , δ K ) is a continuousmap, hence a closed set of ( X, κ ( δ )). Thus, k A ≤ δ ( − , A ). On the other hand, if λ is a closedset of ( X, κ ( δ )) satisfying k A ≤ λ , then for all x ∈ X , δ ( x, A ) ≤ δ K ( λ ( x ) , λ ( A )) = inf λ ( A ) → λ ( x ) ≤ λ ( x ) , hence δ ( − , A ) ≤ λ . Therefore, k A = δ ( − , A ).From this fact one immediately infers that for each Q -approach space ( X, δ ), ζ ◦ κ ( δ ) = δ ,hence κ is right inverse to ζ . To see that κ is left adjoint to ζ , we show that for each Q -cotopological space ( Y, τ ), if f : ( X, δ ) −→ ( Y, ζ ( τ )) is a continuous map (between Q -approachspaces), then f : ( X, κ ( δ )) −→ ( Y, τ ) is a continuous map (between Q -cotopological spaces).That means, for each closed set λ of ( Y, τ ), the map λ ◦ f : ( X, δ ) −→ ( Q , δ K ) is continuous.Let x be an element and A be a subset of X . Without loss of generality, we assume that A = ∅ . Let p be the meet of λ ◦ f ( A ) in Q . Then p → λ is a closed set of ( Y, τ ) with( p → λ )( f ( a )) ≥ k for all a ∈ A , so the closure of k f ( A ) in ( Y, τ ) is contained in p → λ , i.e., k f ( A ) ≤ p → λ . Therefore, δ ( x, A ) ≤ ζ ( τ )( f ( x ) , f ( A ))= k f ( A ) ( f ( x )) ≤ ( p → λ )( f ( x ))= inf λ ◦ f ( A ) → λ ◦ f ( x )= δ K ( λ ◦ f ( x ) , λ ◦ f ( A )) , which shows that λ ◦ f : ( X, δ ) −→ ( Q , δ K ) is continuous, as desired.If the quantale Q is the interval [0 ,
1] together with a continuous t-norm, there is anice characterization of the image of the functor κ , it consists precisely of the strong Q -cotopological spaces.We say that a Q -cotopology τ on a set X is strong [51] if it is co-stratified in the sensethat(C5) p & φ ∈ τ for all p ∈ Q and φ ∈ τ .In other words, a strong Q -cotopology is a quasi- Q -cotopology that is both stratified andco-stratified. 13 roposition 4.7. ([7]) Let ( X, τ ) be a quasi- Q -cotopological space. Then ( X, τ ) is a strong Q -cotopological space if and only if the closure operator satisfies (cl6) p & µ = p & µ for all p ∈ Q and µ ∈ Q X . If Q is the interval [0 ,
1] endowed with a continuous t-norm, strong Q -cotopological spacesare, in essence, fuzzy T -neighborhood spaces in [39, 17]. In particular, if Q is the interval[0 ,
1] endowed with the G¨odel t-norm, i.e., Q = ([0 , , ∧ , Q -cotopologicalspace is strong if and only if it is a fuzzy neighborhood space in the sense of Lowen [34]. Proposition 4.8.
Let Q = ([0 , , & , with & being a continuous t-norm. Then for each Q -approach space ( X, δ ) , κ ( X, δ ) is a strong Q -cotopological space. Moreover, every strong Q -cotopological space arises in this way; explicitly, for each strong Q -cotopological space ( X, τ ) , τ = κ ◦ ζ ( τ ) . Therefore, the functor ζ restricts to an isomorphism between the subcategoryconsisting of strong Q -cotopological spaces and the category of Q -approach spaces, with κ beingits inverse. A lemma first.
Lemma 4.9.
Let Q = ([0 , , & , with & being a continuous t-norm. Then for each strong Q -cotopological space ( X, τ ) and each λ ∈ [0 , X , the closure of λ is given by λ = _ p ∈ [0 , p & λ [ p ] , where, λ [ p ] : X −→ [0 , , λ [ p ] ( x ) := ( , λ ( x ) ≥ p, , otherwise . Proof.
The conclusion is contained in [17, Theorem 1.3], it is essentially the implication (iii) ⇒ (v) therein. A direct proof is included here for the sake of completeness. For each naturalnumber n ≥
2, since _ ≤ i ≤ n − in & λ [ i/n ] ≤ λ ≤ _ ≤ i ≤ n − i + 1 n & λ [ i/n ] , it follows from (cl6) in Proposition 4.7 that _ ≤ i ≤ n − in & λ [ i/n ] ≤ λ ≤ _ ≤ i ≤ n − i + 1 n & λ [ i/n ] . By uniform continuity of & : [0 , × [0 , −→ [0 , n →∞ _ ≤ i ≤ n − in & λ [ i/n ] = lim n →∞ _ ≤ i ≤ n − i + 1 n & λ [ i/n ] . Therefore, λ = _ p ∈ [0 , p & λ [ p ] , as desired. 14he argument of Lemma 4.9 implies that for each closed set λ of a strong Q -cotopologicalspace ( X, τ ), it holds that λ = ^ n ≥ _ ≤ i ≤ n − i + 1 n & λ [ i/n ] . (4.iii)So, { A | A ⊆ X } is a subbasis of τ . In other words, τ is the coarsest strong Q -cotopologyon X that has all 1 A (closure of 1 A in ( X, τ )), A ⊆ X , as closed sets. Proof of Proposition 4.8.
First, we show that for each Q -approach space ( X, δ ), κ ( X, δ ) is astrong Q -cotopological space. To this end, we check that if λ : ( X, δ ) −→ K is continuous,then so is p & λ for all p ∈ [0 , x and each nonempty subset A of X , δ ( x, A ) ≤ δ K ( λ ( x ) , λ ( A ))= inf λ ( A ) → λ ( x ) ≤ ( p & inf λ ( A )) → ( p & λ ( x ))= inf( p & λ )( A ) → ( p & λ )( x ) (& is continuous)= δ K (( p & λ )( x ) , ( p & λ )( A )) . Next, we show that if (
X, τ ) is a strong Q -cotopological space, then τ = κ ◦ ζ ( τ ); that is,for each λ : X −→ [0 , λ is a closed set of ( X, τ ) if and only if λ : ( X, ζ ( τ )) −→ K is a continuous map.Assume that λ is a closed set of ( X, τ ). Let x be a point and A be a nonempty subset of X . Set p = inf λ ( A ) . Then p A ≤ λ and, by (cl6) in Proposition 4.7, p & 1 A ( x ) = p A ( x ) ≤ λ ( x ) . Hence ζ ( τ )( x, A ) = 1 A ( x ) ≤ p → λ ( x ) = δ K ( λ ( x ) , λ ( A )) , which shows that λ : ( X, ζ ( τ )) −→ K is continuous.Conversely, assume that λ : ( X, ζ ( τ )) −→ K is continuous. Then for each point x andeach nonempty subset A of X ,1 A ( x ) = ζ ( τ )( x, A ) ≤ δ K ( λ ( x ) , λ ( A )) = inf λ ( A ) → λ ( x ) , hence inf λ ( A ) & 1 A ( x ) ≤ λ ( x ) . For each p ∈ [0 , A = { a ∈ X | λ ( a ) ≥ p } gives that p & λ [ p ] ≤ λ. Consequently, _ p ∈ [0 , p & λ [ p ] ≤ λ, which implies that λ is a closed set by the above lemma.15 heorem 4.10. Let & be a continuous t-norm and let Q = ([0 , , & , . Then a separated Q -approach space is injective if and only if it is a retract of some power of the space K .Proof. Since the correspondence (
X, δ ) ( X, κ ( δ )) is an isomorphism between the categoryof Q -approach spaces and that of strong Q -cotopological spaces, it suffices to check that inthe category of strong Q -cotopological spaces, a T space is injective if and only if it is aretract of some power of ([0 , , κ ( δ K )).Since for each nonempty subset A of [0 , δ K ( − , A ) = inf A → id , it follows from equation (4.iii) that κ ( δ K ) is the strong Q -cotopology on [0 ,
1] generated, asa subbasis, by the identity map. Then, by the argument of Proposition 4.8 one sees that amap λ : X −→ [0 ,
1] is a closed set of a strong Q -cotopological space ( X, τ ) if and only if λ : ( X, τ ) −→ ([0 , , κ ( δ K )) is continuous. With help of this fact, one readily checks thatin the category of strong Q -cotopological spaces, a T space is injective if and only if it is aretract of some power of ([0 , , κ ( δ K )). Standing Assumption and Convention.
From now on, we always assume that Q is theinterval [0 ,
1] equipped with a continuous t-norm &, though some of the results hold for amore general quantale. The reason is that, for such a quantale, injective and separated Q -approach spaces are precisely retracts of powers of the space K by Theorem 4.10. So, in thesubsequent sections we speak of [0 , , , Q -approach space, Q -order and complete Q -lattice, respectively. [0 , -lattices First, we recall some basic notions in domain theory [13].Let X be a partially ordered set. Write Idl X for the set of all ideals (= directed lowersets) of X endowed with the inclusion order. Since for each x ∈ X , the principal lower set ↓ x is an ideal, we obtain a map y : X −→ Idl
X, x
7→ ↓ x. We say that • X is a directed complete partially ordered set, a dcpo for short, if y : X −→ Idl X has a left adjoint, which necessarily maps each ideal to its join and is denoted bysup : Idl X −→ X . • X is a domain if X is a dcpo and the left adjoint of y : X −→ Idl X also has a leftadjoint. In other words, X is a domain if there is a string of adjunctions ։ ⊣ sup ⊣ y : X −→ Idl X. • X is a continuous lattice if it is at the same time a complete lattice and a domain.Given a dcpo X and two elements x and y , we say that x is way below y , in symbols x ≪ y , if for every directed set D ⊆ X , y ≤ W D implies that x ≤ d for some d ∈ D . It is16nown that a dcpo X is a domain if and only if for all a ∈ X , ։ a := { x ∈ X | x ≪ a } is anideal and has a as a join.A map f : X −→ Y between partially ordered sets is Scott continuous if it preservesdirected joins.Let X be a [0 , { x i } i of X is said to be forward Cauchy [5, 10, 14] if _ i ^ i ≤ j ≤ l X ( x j , x l ) = 1 . A weight φ of X is said to be forward Cauchy if φ = _ i ^ i ≤ j X ( − , x j )for some forward Cauchy net { x i } i of X .Forward Cauchy weights are an analogue of ideals in order theory. A comparative studyof extensions of the notion of ideals to the quantale-valued context can be found in [30].There is a nice characterization of forward Cauchy weights, which says that such weights areprecisely the inhabited and irreducible ones. Proposition 5.1. ([30, Theorem 3.13])
Let X be a [0 , -ordered set. Then a weight φ of X is forward Cauchy if, and only if, (1) φ is inhabited in the sense that W x ∈ X φ ( x ) = 1 ; and (2) φ is irreducible in the sense that sub X ( φ, φ ∨ φ ) = sub X ( φ, φ ) ∨ sub X ( φ, φ ) for all weights φ , φ of X . For each [0 , X , we write D X for the [0 , , P X . Since for each [0 , f : X −→ Y and each forward Cauchy weight φ of X , f → ( φ ) is a forward Cauchy weight of Y , the correspondence f : X −→ Y f → : D X −→ D Y defines a functor D : [0 , Ord −→ [0 , Ord which is a subfunctor of P : [0 , Ord −→ [0 , Ord . Furthermore, it is known, see e.g.[10, 29], that D is a saturated class of weights, hence gives rise to a submonad D = ( D , m , y )of the monad P = ( P , m , y ) . That means, the subfunctor D satisfies the following requirements:17i) For each [0 , X and each x ∈ X , y X ( x ) ∈ D X . So y is also a naturaltransformation from the identity functor to D .(ii) D is closed under multiplication in the sense that for each [0 , X and eachΦ ∈ DD X , m X ◦ ( ε ∗ ε ) X (Φ) ∈ D X, where ε is the inclusion transformation of D in P and ε ∗ ε stands for the horizontalcomposite of ε with itself. So m determines a natural transformation D −→ D , whichis also denoted by m .Let a be an element and { x i } i be a forward Cauchy net of a [0 , X . We saythat a is a Yoneda limit [5, 10, 14] of { x i } i if X ( a, y ) = _ i ^ i ≤ j X ( x j , y )for all y ∈ X . Yoneda limits of a forward Cauchy net, if exist, are unique up to isomorphism.We say that a [0 , X is Yoneda complete [5, 14] if every forward Cauchy net of X has a Yoneda limit.The following conclusion relates Yoneda limits of forward Cauchy nets to suprema offorward Cauchy weights. Proposition 5.2. ([10, Lemma 46])
An element a in a [0 , -ordered set X is a Yoneda limitof a forward Cauchy net { x i } i if and only if a is a supremum of the forward Cauchy weight W i V i ≤ j X ( − , x j ) . Corollary 5.3. A [0 , -ordered set X is Yoneda complete if and only if every forward Cauchyweight of X has a supremum, if and only if the map y : X −→ D X has a left adjoint sup : D X −→ X which necessarily maps each forward Cauchy weight to its supremum. Yoneda completeness is a counterpart, in the realm of [0 , , f : X −→ Y is Yoneda continuous if f preservesYoneda limits of forward Cauchy nets [5, 10, 14]; that is, if x is a Yoneda limit of a forwardCauchy net { x i } i , then f ( x ) is a Yoneda limit of the forward Cauchy net { f ( x i ) } i . It followsfrom Proposition 5.2 that a [0 , , Ord ↑ for the category of [0 , , DCPO of [0 , Ord ↑ composed of separated and Yoneda complete [0 , D , m , y ) , and it is an analogue in the[0 , DCPO of directed complete partially ordered sets andScott continuous maps.
Proposition 5.4.
The underlying ordered set of each Yoneda complete [0 , -ordered set isdirected complete. roof. The conclusion is essentially [32, Proposition 4.5] when & is isomorphic to the productt-norm. Here we present a simple proof for the general case. Let X be a Yoneda complete[0 , D be a directed subset of X , viewed as a forward Cauchy net in X . By assumption, the forward Cauchy net D has a Yoneda limit, say a . By Proposition5.2, a is a supremum of the weight W x ∈ D V y ≥ x X ( − , y ). Since V y ≥ x X ( − , y ) = X ( − , x ) forall x ∈ D , it follows that a is a supremum of the weight W x ∈ D X ( − , x ). From this one easilydeduces that a is a join of D in X .The above argument makes use of the fact that for each directed set D of X , the weight W d ∈ D X ( − , d ) is forward Cauchy. The following conclusion says that for a complete [0 , X , all forward Cauchy weights of X are of this form. Proposition 5.5. ([29, Proposition 4.8])
Let X be a complete [0 , -lattice and let φ be aforward Cauchy weight of X . Then Λ( φ ) = { x ∈ X | φ ( x ) = 1 } is an ideal of X and φ = _ x ∈ Λ( φ ) X ( − , x ) . Furthermore, sup φ is the join of Λ( φ ) in X . Corollary 5.6. ([29, Corollary 4.9]) A [0 , -order-preserving map f : X −→ Y betweencomplete [0 , -lattices is Yoneda continuous if and only if f : X −→ Y is Scott continuous. Definition 5.7.
Let X be a separated [0 , X is a [0 , , d ⊣ sup ⊣ y : X −→ D X ;(2) X is a continuous [0 , , , Proposition 5.8.
The underlying order of a continuous [0 , -lattice is a continuous lattice;that is, if X is a continuous [0 , -lattice, then X is a continuous lattice.Proof. Contained in [29, Proposition 4.11].
Proposition 5.9.
Let X be a complete [0 , -lattice such that X is a continuous lattice.Then the following conditions are equivalent: (1) X is a continuous [0 , -lattice. (2) For each x ∈ X and each forward Cauchy weight φ of X , X ( x, sup φ ) = ^ y ≪ x φ ( y ) , where ≪ denotes the way below relation in X . The map d : X −→ D X, d ( x ) = _ y ≪ x X ( − , y ) preserves [0 , -order, where ≪ denotes the way below relation in X .Proof. This proposition is a slightly different formulation of [29, Proposition 4.11]. We check(3) ⇒ (1) here. By Proposition 5.5 one sees that for each x ∈ X , W y ≪ x X ( − , y ) is thesmallest forward Cauchy weight of X (under pointwise order) that has x as supremum, so, d : X −→ ( D X ) is left adjoint to sup : ( D X ) −→ X . Hence, d : X −→ D X is left adjointto sup : D X −→ X by Theorem 2.7. Example 5.10.
For each continuous t-norm &, the [0 , , , α R ) is a continu-ous [0 , X for the [0 , , , α R ). By Proposition 5.9, it sufficesto check that the map d : X −→ D X, d ( t )( x ) = ( t → x, t = 1 , W b>t ( b → x ) , t < , t < s < s → t ≤ ^ x ∈ [0 , (cid:16) _ a>t ( a → x ) → _ b>s ( b → x ) (cid:17) . (5.iv)For each a > t , let b = ( s → t ) → a . By continuity of & one readily verifies that b > s . Then,the inequality (5.iv) follows from the fact that for each x ∈ [0 , s → t ) &( a → x ) ≤ (( s → t ) → a ) → x = b → x. Example 5.11.
For a continuous t-norm &, the [0 , , , α L ) is a continuous[0 , X for the [0 , , , α L ). By Proposition 5.9, it suffices to show thatthe map d : X −→ D X, d ( t )( x ) = ( x → , t = 0 , W b In this example, we assume that the continuous t-norm & satisfies thecondition (S). Let X = [0 , ∪ {∞} and let X ( x, y ) = x → y, x, y ∈ [0 , , , x = y = ∞ , , x ∈ [0 , , y = ∞ ,y, x = ∞ , y ∈ [0 , . Since forward Cauchy nets in X are, essentially, the forward Cauchy nets in ([0 , , α L )plus constant nets with value ∞ , it is easily verified that X is a separated and Yonedacomplete [0 , d : X −→ D X, d ( x ) = X ( − , ∞ ) , x = ∞ ,X ( − , , x = 0 , W y 1] defined by w ( x, y ) = ^ φ ∈D X X ( y, sup φ ) → φ ( x ) . Some basic properties of the way below [0 , w : for all x, y, z, u ∈ X ,(1) w ( x, y ) ≤ X ( x, y );(2) w ( y, z ) & X ( x, y ) ≤ w ( x, z ), in particular, w ( − , z ) is a weight of X ;(3) X ( z, u ) & w ( y, z ) ≤ w ( y, u ), in particular, w ( y, − ) preserves [0 , Proposition 5.13. ([21, 22, 47]) Let X be a separated and Yoneda complete [0 , -ordered set.Then X is a [0 , -domain if and only if for all x ∈ X , the weight w ( − , x ) is forward Cauchywith x being a supremum. In this case, the left adjoint d : X −→ D X of sup : D X −→ X isgiven by d ( x ) = w ( − , x ) . orollary 5.14. Let X be a continuous [0 , -lattice. Then for all x, y ∈ X , w ( x, y ) = _ a ≪ y X ( x, a ) = _ a ≪ y w ( x, a ) , where ≪ refers to the way below relation in X .Proof. The first equality follows immediately from Proposition 5.9 and Proposition 5.13. Tosee the second equality, assume that a ≪ y in X . Pick some c ∈ X such that a ≪ c ≪ y in X . Then X ( x, a ) ≤ w ( x, c ) by the first equality, and consequently, _ a ≪ y X ( x, a ) ≤ _ a ≪ y w ( x, a ) . The converse inequality is trivial.The following conclusion, known as the interpolation property, has appeared in differentcontexts in the literature, see e.g. [21, 22, 45], a direct verification is included here for thesake of completeness. Proposition 5.15. The way below [0 , -relation in a [0 , -domain X has the interpolationproperty in the sense that w ( x, y ) = _ z ∈ X w ( z, y ) & w ( x, z ) . Proof. Since X is a [0 , d : X −→ D X, y w ( − , y )is a left adjoint, it follows that for each y ∈ X , d ( y ) = d (sup w ( − , y )) = sup D X d → ( w ( − , y )) . Since for each φ ∈ D X , d → ( w ( − , y ))( φ ) = _ z ∈ X w ( z, y ) & D X ( φ, d ( z )) , it follows that for all x, y ∈ X , w ( x, y ) = d ( y )( x )= _ φ ∈D X (cid:16)(cid:16) _ z ∈ X w ( z, y ) & D X ( φ, w ( − , z )) (cid:17) & φ ( x ) (cid:17) = _ z ∈ X w ( z, y ) & w ( x, z ) . The following definition is a direct extension of that for generalized metric spaces (i.e.,ordered sets valued in Lawvere’s quantale) in [5, 14]. Definition 5.16. An element a of a [0 , X is called compact if X ( a, − ) : X −→ ([0 , , α L )is Yoneda continuous. A Yoneda complete [0 , emma 5.17. Let X be a Yoneda complete [0 , -ordered set and let a be an element of X .Then the following conditions are equivalent: (1) a is compact. (2) For each forward Cauchy weight φ of X , X ( a, sup φ ) = φ ( a ) . (3) w ( − , a ) = X ( − , a ) . Proposition 5.18. Every separated algebraic [0 , -ordered set is a [0 , -domain. Such a [0 , -domain is called an algebraic [0 , -domain.Proof. Similar to the proof of [33, Proposition 5.4]. Corollary 5.19. For a continuous t-norm & on [0 , , the following conditions are equivalent: (1) & is Archimedean; that is ([0 , , & , has no idempotent element other than and . (2) The complete [0 , -lattice ([0 , , α L ) is algebraic. (3) The complete [0 , -lattice ([0 , , α R ) is algebraic.Proof. This follows from a combination of Example 5.10, Example 5.11, Lemma 5.17 andProposition 5.18, details are left to the reader.We write [0 , ConLat for the category with • objets: continuous [0 , • morphisms: Yoneda continuous right adjoints.If the continuous t-norm & satisfies the condition (S), then [0 , ConLat is monadic overthe category of sets (see Theorem 5.23 below). In particular, [0 , ConLat is a completecategory. For an arbitrary continuous t-norm &, we do not know whether [0 , ConLat ismonadic over sets, however, it is still a complete category.First of all, we note that from Proposition 2.10 and [29, Proposition 3.4] it follows thatevery retract of a [0 , , , Ord ↑ isa [0 , , Lemma 5.20. Suppose that X, Y are separated [0 , -ordered sets, and that g : X −→ Y is aYoneda continuous right adjoint. (1) If g is surjective and X is a continuous [0 , -lattice, then so is Y . (2) If g is injective and Y is a continuous [0 , -lattice, then so is X .Proof. (1) Let f : Y −→ X be the left adjoint of g . Then g ◦ f = 1 Y . Thus, as a retract of X in [0 , Ord ↑ , Y is a continuous [0 , f : Y −→ X be the left adjoint of g . Then f ◦ g = 1 X . Thus, as a retract of Y in[0 , Ord ↑ , X is a continuous [0 , , k : X −→ X is a kernel operator if k = k and k ( x ) ≤ x in X for all x ∈ X . We leave it to the reader to check that if k : X −→ X is akernel operator, then k : X −→ k ( X ) is right adjoint to the inclusion k ( X ) −→ X and k ( X )is a retract of X in [0 , Ord . In particular, if X is a continuous [0 , k : X −→ X is a Yoneda continuous kernel operator, then k : X −→ k ( X ) is a Yoneda continuous rightadjoint, hence k ( X ) is a continuous [0 , Proposition 5.21. The category [0 , - ConLat is complete.Proof. We check that [0 , ConLat has equalizers and products. Given a parallel pair ofmorphisms f, g : X −→ Y in [0 , ConLat , let Z = { x ∈ X | f ( x ) = g ( x ) } . The inclusion i : Z −→ X is clearly Yoneda continuous. Since i : Z −→ X is an equalizer ofthe pair f, g : X −→ Y in the category of continuous lattices, it is a right adjoint betweenpartially ordered sets. Thus, by Theorem 2.7 we obtain that i : Z −→ X is a Yonedacontinuous right adjoint. Therefore, Z is a continuous [0 , i : Z −→ X is an equalizer of f, g : X −→ Y in [0 , ConLat .To see that [0 , ConLat has products, we check that for each family { X i } i ∈ I of continuous[0 , Q i X i is a continuous [0 , x of X and each ideal D of X , X ( x, _ D ) = ^ y ≪ x _ d ∈ D X ( y, d ) . (5.v)So, we only need to check that for each ~x = ( x i ) i ∈ Q i X i and each ideal D in the underlyingorder of Q i X i , ^ i X i ( x i , a i ) = ^ ~y ≪ ~x _ ~d ∈ D ^ i X i ( y i , d i ) , (5.vi)where ~a = ( a i ) i is the join of D in the underlying order of Q i X i .Assume that r < V i X i ( x i , a i ) and ~y ≪ ~x . We are to find an element ~c of D such r ≤ V i X i ( y i , c i ), which proves the inequality ≤ in (5.vi). Write ⊥ for the bottom element inthe underlying order of each X i . Since ~y ≪ ~x , then y i ≪ x i for all i ∈ I and y i = ⊥ for allbut a finite number of i . Put c i = ⊥ if y i = ⊥ . Now we define c i when y i = ⊥ . Since X i is acontinuous [0 , X i ( x i , a i ) ≤ _ b i ≪ a i X i ( y i , b i ) , which implies that there is some b i ≪ a i such that r ≤ X i ( y i , b i ). Put c i to be this b i . Then ~c = ( c i ) i is an element of D that satisfies the requirement.For the converse inequality, we note at first that for each i , a i is the join of the directedset { d i } ~d ∈ D in the underlying order of X i . Since X i is a continuous [0 , X i ( x i , a i ) = ^ z ≪ x i _ ~d ∈ D X i ( z, d i ) . z ≪ x i , define an element ~y = ( y j ) j of the product Q j X j by letting y i = z and y j = ⊥ for j = i . Then ~y ≪ ~x and for all ~d ∈ D , ^ j X j ( y j , d j ) = X i ( z, d i ) , and consequently, ^ i X i ( x i , a i ) = ^ i ^ z ≪ x i _ ~d ∈ D X i ( z, d i ) ≥ ^ ~y ≪ ~x _ ~d ∈ D ^ i X i ( y i , d i ) . Proposition 5.22. For a continuous t-norm & satisfying the condition (S), a Q -ordered set X is a continuous [0 , -lattice if and only if it is a retract of some power of ([0 , , α L ) in thecategory [0 , - Ord ↑ .Proof. Necessity follows from Example 5.11, Proposition 5.21 and the fact that every retractof a continuous [0 , , Ord ↑ is a continuous [0 , X is a continuous [0 , X is a retract of D X in category[0 , DCPO . Since the inclusion D X −→ P X is Yoneda continuous, then D X is a retract of P X in [0 , DCPO by [29, Theorem 6.4 (6)]. Finally, since P X is a retract of ([0 , , α L ) X inthe category [0 , Sup , it follows that X is a retract of ([0 , , α L ) X in [0 , Ord ↑ .It is well-known that the category of continuous lattices is monadic over sets, see e.g.[8, 49]. This is still true for continuous [0 , Theorem 5.23. If & satisfies the condition (S), then the category [0 , - ConLat is monadicover the category of sets.Proof. Since & satisfies the condition (S), it follows from Corollary 5.6 and Theorem 6.4 in [29]that the forgetful functor [0 , ConLat −→ [0 , Ord is monadic, hence a right adjoint. It isclear that the forgetful functor [0 , Ord −→ Set is also a right adjoint, so the forgetful functor[0 , ConLat −→ Set , as a composite of right adjoints, is a right adjoint. It remains, by Beck’stheorem (see e.g. [38, page 151]), to check that the forgetful functor [0 , ConLat −→ Set creates split coequalizers. This is contained in Proposition 5.25 below.Let X be a [0 , y ∈ X and let p ∈ [0 , cotensor of p and y [44,page 288] is an element p y of X such that for all x ∈ X , X ( x, p y ) = p → X ( x, y ) . Some useful facts about cotensors are listed below, which can be found in [44] and [28]. • If X is a complete [0 , p y exist. • Every right adjoint f preserves cotensor; that is, f ( p y ) is a cotensor of p and f ( y ). • A map f : X −→ Y between complete [0 , , f : X −→ Y preserves order and f ( p x ) ≤ p f ( x ) for all x ∈ X and p ∈ [0 , Lemma 5.24. Suppose that X be a complete [0 , -lattice; suppose that R is a relation on X subject to the following conditions: R is closed w.r.t. directed joins in X × X ; (ii) R is closed w.r.t. meets in X × X ; (iii) If ( x, y ) ∈ R , then ( p x, p y ) ∈ R for all p ∈ [0 , .Then, the map k : X −→ X, k ( x ) = ^ { y | ( x, y ) ∈ R } is a Yoneda continuous kernel operator.Proof. Routine. Proposition 5.25. The forgetful functor [0 , - ConLat −→ Set creates split coequalizers.Proof. Let f, g : X / / Y be a parallel pair of morphisms in [0 , ConLat ; and let h : Y −→ Z be a split coequalizer of f, g in Set . By definition there exist morphisms Z i / / Y j / / X in Set such that h ◦ f = h ◦ g, f ◦ j = id , h ◦ i = id , g ◦ j = i ◦ h. Let R = { ( y , y ) ∈ Y × Y | h ( y ) = h ( y ) } . It is not hard to check that ( y , y ) ∈ R if and only if there is some ( x , x ) ∈ X × X such that g ( x ) = g ( x ), y = f ( x ) and y = f ( x ). With help of this fact, one readily verifies that R satisfies the conditions (i)-(iii) in Lemma 5.24, hence R determines a Yoneda continuouskernel operator k : Y −→ Y . Since k ( Y ) is a continuous [0 , Z , Z can be made into a continuous [0 , h : Y −→ Z is aYoneda continuous right adjoint. This proves that the forgetful functor [0 , ConLat −→ Set creates split coequalizers. [0 , -approach spaces In this section we introduce the notion of Scott [0 , , , , , Definition 6.1. ([46, Definition 4.4]) A weight φ of a [0 , X is Scott closed if,as a [0 , φ : X −→ ([0 , , α R ) is Yoneda continuous. Lemma 6.2. Let X be a [0 , -ordered set; let φ : X −→ ([0 , , α R ) be a [0 , -order-preservingmap. The following conditions are equivalent: (1) φ is a Scott closed weight of X . (2) φ : X −→ ([0 , , α R ) preserves suprema of forward Cauchy weights. (3) For each forward Cauchy weight λ of X , sub X ( λ, φ ) = φ (sup λ ) . (4) For each forward Cauchy weight λ of X , sub X ( λ, φ ) ≤ φ (sup λ ) .Proof. Routine, for example, (2) ⇒ (3) follows from Equation (2.ii).26 roposition 6.3. ([30, Proposition 5.9]) For each [0 , -ordered set ( X, α ) , the set σ ( α ) ofScott closed weights of ( X, α ) is a strong [0 , -cotopology on X . By Proposition 4.8, the correspondence ( X, τ ) ( X, ζ ( τ )) is an isomorphism between thecategories of strong [0 , , , X, τ ) with the [0 , X, ζ ( τ )), viewingthe [0 , τ and the [0 , ζ ( τ ) as different facets of the sameobject, as in the theory of approach spaces [36, 37]. In particular, for each [0 , X, α ), we identify the strong [0 , X, σ ( α )) with the [0 , X, ζ ( σ ( α ))) and call σ ( α ) and ζ ( σ ( α )) the Scott [0 , -cotopology and the Scott [0 , -approach structure of ( X, α ), respectively.Since Lawvere’s quantale ([0 , ∞ ] op , + , 0) is isomorphic to the quantale ([0 , , & P , , , X , we writeΣ( X, α ) , or simply Σ X if α is clear from the context, for the [0 , X, σ ( α )) and/orthe [0 , X, ζ ( σ ( α ))). Example 6.4. ([46, Lemma 4.13]) For each element a of a [0 , X ( − , a ) is Scott closed. This is because that for each forward Cauchy weight λ of X ,sub X ( λ, X ( − , a )) = X (sup λ, a )by definition of sup λ . It is clear that X ( − , a ) is the closure of 1 a in Σ X .In particular, for each r ∈ [0 , → r is a Scott closed weight of ([0 , , α L ), henceid → r : ([0 , , α L ) −→ ([0 , , α R )is Yoneda continuous.If f : X −→ Y is Yoneda continuous, then f : Σ X −→ Σ Y is continuous. Hence weobtain a functor: Σ : [0 , Ord ↑ −→ [0 , App . Proposition 6.5. The functor Σ is full; that is, a map f : X −→ Y between [0 , -orderedsets is Yoneda continuous if and only if f : Σ X −→ Σ Y is continuous.Proof. This is, in essence, Proposition 4.15 in [46].It is easily verified that for each [0 , X , the specialization [0 , , X coincides with that of X ; that is, ΩΣ X = X . In particular, thefunctor Σ : [0 , Ord ↑ −→ [0 , App is a full embedding. Example 6.6. Since a map φ : ([0 , , α R ) −→ ([0 , , α R ) is Yoneda continuous if andonly if it is right continuous and preserves [0 , , , α R ) = K . Thus, the Scott [0 , , , α R ) is the strong [0 , , 1] generated by the identity map as a subbasis, giving a simple proof of the assertionof [30, Example 5.10]. 27 closed set φ of a [0 , X is called irreducible if W x ∈ X φ ( x ) = 1 andsub X ( φ, ψ ∨ ψ ) = sub X ( φ, ψ ) ∨ sub X ( φ, ψ )for all closed sets ψ , ψ of X . A [0 , X is called sober [52] if for eachirreducible closed set φ of X , there is a unique x ∈ X such that φ is the closure of 1 x .We say that a [0 , X, δ ) is sober if the [0 , X, κ ( δ ))is sober. If the continuous t-norm & is isomorphic to the product t-norm, then this notion is,in essence, that of sober approach spaces in [4, 32] postulated in terms of Lawvere’s quantale.From Proposition 5.1 and [52, Proposition 3.10] one easily deduces that the specialization[0 , , , , Lemma 6.7. Let X be a [0 , -domain and let φ be a weight of X . Then the closure of φ in Σ X is given by φ ( a ) = sub X ( w ( − , a ) , φ ) for all a ∈ X , where w is the way below [0 , -relationof X .Proof. First, we show that φ is Scott closed if and only if for all a ∈ X , φ ( a ) = sub X ( w ( − , a ) , φ ) . Necessity follows from Lemma 6.2 (3) and that w ( − , a ) is a forward Cauchy weight with a being a supremum by Proposition 5.13. For sufficiency, we check that for each forwardCauchy weight λ , sub X ( λ, φ ) ≤ φ (sup λ ). Let a = sup λ . Since w ( − , a ) is a forward Cauchyweight and w ( − , a ) ≤ λ , then sub X ( λ, φ ) ≤ sub X ( w ( − , a ) , φ ) = φ (sup λ ), as desired.Next, we prove the conclusion. Write ψ for the map X −→ [0 , 1] defined by ψ ( a ) = sub X ( w ( − , a ) , φ ) . It is easily verified that (i) ψ is a weight of X ; (ii) φ ≤ ψ ; and (iii) ψ ≤ λ for any Scott closedweight λ that majorizes φ . To finish the proof, it remains to check that ψ is Scott closed.By the interpolation property of the way below [0 , w , one easily verifies that ψ satisfies that ψ ( a ) = sub X ( w ( − , a ) , ψ ) for all a ∈ X , hence it is Scott closed. Proposition 6.8. For each [0 , -domain X , the space Σ X is sober.Proof. Let λ be an irreducible closed set of Σ X . We have to show that there is a uniqueelement b of X such that λ is the closure of 1 b , i.e., λ = X ( − , b ). Uniqueness is obvious since X is separated. Now we prove the existence. Let ⇓ λ = _ a ∈ X λ ( a ) & w ( − , a ) , where w is the way below [0 , X . We show in two steps that ⇓ λ is a forwardCauchy weight of X and its supremum satisfies the requirement. Step 1 . We use the characterization in Proposition 5.1 to show that ⇓ λ is a forwardCauchy weight of X .For each a ∈ X , since w ( − , a ) is a forward Cauchy weight, then W x ∈ X w ( x, a ) = 1 byProposition 5.1. Thus, _ x ∈ X ⇓ λ ( x ) = _ a ∈ X _ x ∈ X λ ( a ) & w ( x, a ) = _ a ∈ X λ ( a ) = 1 , ⇓ λ is inhabited.To see that ⇓ λ is irreducible, first we show that for each weight φ of X ,sub X ( ⇓ λ, φ ) = sub X ( λ, φ ) . We calculate: sub X ( ⇓ λ, φ ) = ^ a ∈ X ( λ ( a ) → sub X ( w ( − , a ) , φ ))= ^ a ∈ X ( λ ( a ) → φ ( a ))= sub X ( λ, φ ) . Then, for any weights φ , φ of X , we havesub X ( ⇓ λ, φ ∨ φ ) = sub X ( λ, φ ∨ φ )= sub X ( λ, φ ∨ φ )= sub X ( λ, φ ) ∨ sub X ( λ, φ )= sub X ( ⇓ λ, φ ) ∨ sub X ( ⇓ λ, φ ) , hence ⇓ λ is irreducible. Step 2 . Since ⇓ λ is a forward Cauchy weight, it has a supremum, say b . We claim that b satisfies the requirement.On one hand, since λ is Scott closed, then 1 = sub X ( ⇓ λ, λ ) = λ (sup ⇓ λ ) = λ ( b ) , hence X ( − , b ) ≤ λ . On the other hand, since for each a ∈ X ,sub X ( w ( − , a ) , ⇓ λ ) ≥ sub X ( w ( − , a ) , λ ( a ) & w ( − , a )) ≥ λ ( a ) , it follows from Lemma 6.7 that λ is the closure of ⇓ λ , hence λ ≤ X ( − , b ) because X ( − , b ) isScott closed and contains ⇓ λ .Now we present the main result in this paper. Theorem 6.9. If X is a separated and injective [0 , -approach space, then Ω( X ) is a con-tinuous [0 , -lattice and X = ΣΩ( X ) . Before proving this theorem, we prove four lemmas first. Lemma 6.10. Let X be a [0 , -domain. For each x ∈ X , w ( x, − ) : X −→ ([0 , , α L ) isYoneda continuous.Proof. Since w ( x, − ) preserves [0 , w ( x, − ) → ( φ ) ≥ w ( x, sup φ )for each forward Cauchy weight φ of X . Since w ( − , sup φ ) ≤ φ , thensup w ( x, − ) → ( φ ) ≥ sup w ( x, − ) → ( w ( − , sup φ ))= _ z ∈ X w ( z, sup φ ) & w ( x, z ) (Equation (2.i))= w ( x, sup φ ) , (Proposition 5.15)which completes the proof. 29 emma 6.11. Let X be a [0 , -domain. Then { w ( x, − ) → r | x ∈ X, r ∈ [0 , } is a subbasisfor the closed sets of Σ X .Proof. Since id → r : ([0 , , α L ) −→ ([0 , , α R ) is Yoneda continuous by Example 6.4 and w ( x, − ) : X −→ ([0 , , α L ) is Yoneda continuous by Lemma 6.10, it follows that for each r ∈ [0 , w ( x, − ) → r , a composite of w ( x, − ) and id → r , is Yoneda continuous,hence a Scott closed weight of X .Now we show that φ ( y ) = ^ x ∈ X w ( x, y ) → φ ( x )for each Scott closed weight φ and each y ∈ X , which entails the conclusion.Since φ : X −→ ([0 , , α R ) is Yoneda continuous, it follows that φ ( y ) = φ (sup w ( − , y ))= sup φ → ( w ( − , y ))= sub X ( w ( − , y ) , φ ) (Equation (2.ii))= ^ x ∈ X w ( x, y ) → φ ( x ) . The proof is completed. Lemma 6.12. Let X be a continuous [0 , -lattice and I be a set. Then for all ~x = ( x i ) i and ~y = ( y i ) i of X I , w ( ~x, ~y ) ≤ V i ∈ I w ( x i , y i ) . If x i is the bottom element of X for all but a finitenumber of i , then w ( ~x, ~y ) = V i ∈ I w ( x i , y i ) .Proof. By Proposition 5.21, X I is a continuous [0 , w ( ~x, ~y ) = _ ~a ≪ ~y X I ( ~x, ~a )For each ~a ≪ ~y and j ∈ I , we have a j ≪ y j , so X I ( ~x, ~a ) ≤ X ( x j , a j ) ≤ _ a ≪ y j X ( x j , a ) = w ( x j , y j ) , which implies that w ( ~x, ~y ) ≤ ^ i ∈ I w ( x i , y i ) . Now suppose there is a finite set J ⊆ I , such that, x i is the bottom element of X for all i ∈ I \ J . Then ^ i ∈ I w ( x i , y i ) = ^ j ∈ J w ( x j , y j ) = ^ j ∈ J _ b j ≪ y j X ( x j , b j ) . For each j ∈ J and b j ≪ y j , let ~c = ( c j ) j ∈ I , where c j = b j if j ∈ J and c j is the bottomelement of X if j / ∈ J . Then ~c ≪ ~y and thus, ^ i ∈ I w ( x i , y i ) ≤ _ ~b ≪ ~y X I ( ~x,~b ) = w ( ~x, ~y ) . emma 6.13. For each set I , Σ(([0 , , α R ) I ) = K I .Proof. We prove a slightly more general result: for each continuous [0 , X and eachset I , Σ( X I ) = (Σ X ) I . By Lemma 6.11, it suffices to check that for each ~x ∈ X I and p ∈ [0 , w ( ~x, − ) → p is a closed set of the product space (Σ X ) I .Given ~y ∈ X I and r < w ( ~x, ~y ), since w ( ~x, ~y ) = _ ~a ≪ ~y w ( ~x, ~a )by Corollary 5.14, there is some ~a r ≪ ~y such that r ≤ w ( ~x, ~a r ). Since ~a r ≪ ~y , we canfind a finite subset J of I , such that a ri is the bottom element of X for i ∈ I \ J . Define λ r~y : X I −→ [0 , 1] by λ r~y ( ~z ) = r ∧ ^ i ∈ J w ( x i , z i ) . It is clear that λ r~y → p is a closed set of (Σ X ) I and r = λ r~y ( ~y ).Next, we show that λ r~y ≤ w ( ~x, − ). Let ~x ′ be the point with x ′ i = x i for each i ∈ J and x ′ i = ⊥ for i / ∈ J . Then for all ~z ∈ X I , λ r~y ( ~z ) = r ∧ w ( ~x ′ , ~z ) (Lemma 6.12)= r ∧ _ ~b ≪ ~z X I ( ~x ′ ,~b )= _ ~b ≪ ~z (cid:0) r ∧ ^ i ∈ J X ( x i , b i ) (cid:1) (Lemma 6.12) ≤ _ ~b ≪ ~z ^ i ∈ I X ( x i , b i )= w ( ~x, ~z ) . Finally, since r = λ r~y ( ~y ), it follows that w ( ~x, − ) = _ { λ r~y | r < w ( ~x, ~y ) , ~y ∈ X I } . Consequently, as a meet of the family of closed sets { λ r~y → p | r < w ( ~x, ~y ) , ~y ∈ X I } ,w ( ~x, − ) → p is a closed set of (Σ X ) I . Remark 6.14. Lemma 6.13 was proved in [33] when & is isomorphic to the product t-norm.The proof therein depends on the fact that for such a t-norm, ([0 , , α R ) is an algebraic[0 , , , α R ) is seldomalgebraic. Proof of Theorem 6.9. By Theorem 4.10, there exist continuous maps s : X −→ K I and r : K I −→ X such that r ◦ s = 1 X . Since Ω( K I ) = ([0 , , α R ) I , it follows from Proposition 2.10that Ω( X ) is a complete [0 , X ) is a [0 , r : ([0 , , α R ) I −→ Ω( X ) and s : Ω( X ) −→ ([0 , , α R ) I are Yoneda continuous,for which we only need to check, by Corollary 5.6, that both r : ([0 , , ≥ ) I −→ Ω( X ) and s : Ω( X ) −→ ([0 , , ≥ ) I are Scott continuous. This can be deduced from the following facts:31 both r : ([0 , , ≥ ) I −→ Ω( X ) and s : Ω( X ) −→ ([0 , , ≥ ) I preserve order; • r ◦ s = 1 X ; and • s ◦ r : ([0 , , ≥ ) I −→ ([0 , , ≥ ) I is Scott continuous, by Proposition 6.5 and Corollary5.6.Since Σ([0 , , α R ) I = K I by Lemma 6.13, it follows that both r : K I −→ ΣΩ( X ) and s : ΣΩ( X ) −→ K I are continuous. Thus, both1 X = r ◦ s : X −→ K I −→ ΣΩ( X )and 1 X = r ◦ s : ΣΩ( X ) −→ K I −→ X are continuous and consequently, X = ΣΩ( X ). Corollary 6.15. Let X be a separated [0 , -approach space. Then X is injective if and onlyif the following conditions are satisfied: (1) Ω( X ) is a retract of some power of ([0 , , α R ) in [0 , - Ord ↑ . (2) X = ΣΩ( X ) .Proof. This follows from a combination of Theorem 6.9 and the following facts: K is injective,Ω( K ) = ([0 , , α R ) and every functor preserves retracts. Lemma 6.16. The [0 , -approach space Σ([0 , , α L ) is injective if and only if & is isomorphicto the Lukasiewicz t-norm.Proof. If & is isomorphic to the Lukasiewicz t-norm, then f : ([0 , , α L ) −→ ([0 , , α R ) , f ( x ) = x → , , α L ) is injective.Next, we show that Σ([0 , , α L ) is not injective if & is not isomorphic to the Lukasiewiczt-norm. If & does not satisfy the condition (S), then ([0 , , α L ) is not continuous by Example5.11, hence Σ([0 , , α L ) is not injective by Theorem 6.9. If & satisfies the condition (S), thenthere exist idempotent elements p, q of the quantale ([0 , , & , 1) such that the restriction of& on [ p, q ] is either isomorphic to the G¨odel t-norm or to the product t-norm.Suppose on the contrary that Σ([0 , , α L ) is injective. Consider the subspace { p, q } of K and the map f : { p, q } −→ Σ([0 , , α L ) , f ( p ) = q , f ( q ) = p. It is clear that f preserves the specialization [0 , , , f is continuous.So, there is a continuous map f : K −→ Σ([0 , , α L ) that extends f . By Lemma 6.5, f : ([0 , , α R ) −→ ([0 , , α L ) is Yoneda continuous, hence f : [0 , −→ [0 , 1] transformsnon-empty meets to joins. Since f ( q ) = p and f is [0 , x ∈ ( p, q ), f ( x ) ≤ α R ( x, q ) → f ( q ) = x → p = p, hence W x>p f ( x ) = p , contradicting that f ( p ) = f ( p ) = q .32 emark 6.17. That Σ([0 , , α L ) is not injective when & is isomorphic to the product t-normis known in [15], Example 4.14 and [33], Example 5.10. Theorem 6.18. Let & be a continuous t-norm that satisfies the condition (S). Then thefollowing statements are equivalent: (1) & is isomorphic to the Lukasiewicz t-norm. (2) Every continuous [0 , -lattice with the Scott [0 , -approach structure is injective.Proof. 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