COUPLED QUANTALES AND A NON-COMMUTATIVE APPROACH TO BITOPOLOGICAL SPACES

The concept of a coupled quantales is introduced as a non-commutative extension of the concept of biframes. Also an approach to non-commutative bitopology is studied. Then an adjunction between the category of coupled quantales and the category of biquantum spaces is established. AMS Subject Classification: 06F07, 06D22, 54E55, 54B35


Introduction
In 1986 C.J. Mulvey [7] proposed the term quantale as a non-commutative extension of the concept of frame ( a complete lattice satisfying the first infinite distributive law of finite meets over arbitrary sups).The purpose was to develop the concept of non-commutative topology, introduced by R. Giles and H. Kummer [5], while providing constructive foundations for the theory of quantum mechanics and non-commutative logic [11].Nowadays, the notion of quantale can boast many areas of application, e. g., in the field of non-commutative topology [8,9,3].Further details about quantales can be found in [10].
In 1989 F. Borceux and G. van den Bossche (c.f.[2])proposed a more general model of non-commutative topology strongly based on the notion of quantale; they define a quantum space as a family of open sets in which the intersection is substituted by a product ⊗, in such a manner that the lattice of open sets is a (right-sided and idempotent) quantale.Also a duality between spatial right-sided idempotent quantales and sober quantum spaces is proposed.Such duality is extended by Höhle [6] to an adjunction based on quantum spaces In this paper we aim to introduce the notions of coupled quantale as a noncommutative extension of the concept of biframe [1].Also, we aim to propose a model of non-commutative bitopology based on the notion of coupled quantale.Then, we will extend the dual adjunction between the category of right-sided idempotent quantales and the category of quantum spaces to one between the category of coupled quantales and the category of biquantum spaces.
By Quant(resp.StQuant), we mean the category of all quantales and quantale morphisms (resp.strong quantale morphisms).StQuant and Quant clearly share the same objects.Definition 2.3.[10]A quantale (Q, ≤, ⊗) is said to be: (1) a unital quantale if whose multiplication ⊗ has an identity element e ∈ L called the unit.USQuant denotes the category all unital quantale together with all quantale morphisms preserving the unit e.
(4) idempotent iff a ⊗ a = a, for all a ∈ L.
By subquantale [10] of of a quantale Q = (Q, ≤, ⊗) we mean subset S ⊆ Q which is closed under the tensor product ⊗ and arbitrary sups.
Remark 2.4.[10] The intersection of two subquantales of quantale Q is also a subquantale of Q.
By a quantic nucleus [10] on a quantale Q, we mean a closure operator c : Lemma 2.5.[6] In any right-sided and idempotent quantale Q the closure operator is a nucleus on Q.
For any isotone maps between f : P −→ Q, between posets, its right adjoint is the map, (which is necessarily unique), f * : In 1989 F. Borceux and G. van den Bossche (c.f.[2]) proposed a more general model of non-commutative topology strongly based on the notion of quantale; they define a quantum space as a set X provided with a family O(X) ⊆ P (X) of open subsets and a multiplication: in such a manner that the lattice of open sets, which called a non-commutative topology, is a right-sided and idempotent quantale as given in the following definition Definition 2.6.[2] Let X be a non-empty set.A non-commutative topology on X is a subset O(X) ⊆ P (X) satisfying the following conditions: (1) O(X) is closed under arbitrary union i.e., the set-inclusion O(X) ֒→ P (X) is a join preserving map.
(3) On O(X) there exists a binary operation: (5) the pair (O(X), ⊗) is a right-sided and idempotent quantale.i.e., If O(X) is a non-commutative topology on X, then the pair (X, O(X)) is called a quantum space An important corollary of the axioms of a non-commutative topology O(X) on X is the following property: A continuous mapping [2] from a quantum space (X, O(X)) to a quantum space A continuous mapping f : X → Y is said to by strict continuous if it satisfies the condition (1) A biquantum space is a triple (X, O 1 (X), O 2 (X)) consisting of a nonempty set X and two non-commutative topologies O 1 (X) and O 2 (X) of subsets of X. ( ) is a function between their underlying sets for which are continuous.
(3) The category of biquantum spaces and bicontinuous maps will be denoted by BiQS.
Example 2.8.Every bitopological space is clearly a biquantum space when defining ) is a function between their underlying sets for which The biquantum spaces and their strict bicontinuous mappings constitute nevertheless an interesting category.
and H ∈ O 1 (X) with the same property.
Then both O 1 (X) and O 2 (X) are a left-sided and idempotent quantales.Thus both are T 0 , but not T 1 .
Definition 2.13.We call a biquantum space (X, Then the following conditions are equivalent: (4) For any two distinct points x, y ∈ X there exist disjoint U, V ∈ O 1 (X) ∪ O 2 (X) such that x ∈ U and y ∈ V . Proof.
(1) ⇒ (2) :Suppose that (X, O 1 (X)) is T 0 and x, y are two distinct points of X.Then there exists U ∈ O 1 (X) containing exactly one of x, y.Without loss of generality we may assume that x ∈ U and y / (2) ⇒ (3) : Suppose that (X, O 2 (X)) is T 0 and x, y are two distinct points of X.
Then there exists U ∈ O 2 (X) containing exactly one of x, y.Without loss of generality we may assume that x ∈ U and y / (3) ⇒ (4) :is obvious.

Duality between BiQS and RSiCQuant
For a fixed Q ∈ |Quant|, it follows, as a consequence of Remark 2.4, that the family of all subquantales of Q, ordered by inclusion, forms a complete lattice, with the meet which is not their set-theoretical union).So, for a subset K ⊆ Q of a quantale Q, the smallest subquantale of Q which contains K is defined to be the subquantale generated by K. Definition 2.16.( The category of coupled quantales ) (2) A map h : Q → P between coupled quantales is a quantale morphism Q 0 → P 0 for which the restrictions h| Q i : Q i → P i are quantale morphisms i.e., h(Q i ) ⊆ P i for i = 1, 2.
(3) The resulting category will be denoted by CQuant.
We refer to Q 0 as the total part of Q, and Q 1 , Q 2 as its first and second parts, respectively.
The restrictions of a coupled quantale map h : Q → P to various parts will be written is said to be: (1) unital iff Q 0 is unital and e belongs to both Q 1 and Q 2 .UnCQuant is the full subcategory of CQuant of all unital coupled quantales.
(3) idempotent iff the total part Q 0 is idempotent i.e., a⊗a = a for all a ∈ Q 0 .
(4) commutative if the operation ⊗ is commutative i.e.,q 1 ⊗ q 2 = q 2 ⊗ q 1 for every q 1 ∈ Q i and q 2 ∈ Q k .ComCQuant is the full subcategory of CQuant of all commutative coupled quantales (5) A biframe [1] is a unital commutative coupled quantale whose multiplication and unit are ∧ and ⊤ respectively and Example 2.18.Let Q = {⊥, a, b, ⊤} be the four Boolean lattice and let Through this paper we use RSiCQuant to denote the category of idempotent and right-sided coupled quantales with coupled quantales morphisms.In this context strictly bicontinuous maps can be viewed as strong homomorphisms w.r.t. the underlying idempotent and right-sided coupled quantales.This observation implies that there exists a functor Ω : BiQS → RSiCQuant op sending a biquantum space to its underlying non-commutative bitopology. i.e., where O 0 (X) = O 1 (X)∨O 2 (X), that is, the coarsest non-commutative topology containing O 1 (X) and O 2 (X).
For the strictly bicontinuous function We show that the functor Ω has a right adjoint.For this purpose we begin with the following observation.
As given in [6], one can associate a non-commutative topology O Q with every right-sided and idempotent quantale Q as follows: Let pt(Q) be the spectrum of Q i.e., the set of all prime elements of Q.Then for every x ∈ Q the set is in general not uniquely be determined by x, but there exists a largest element of Q with this property-namely c(x) where c is the nucleus determined by Eq.( 1).
The complete sublattice Also, the the strong homomorphism takes the form Lemma 2.20.For a strong RSiCQuant-homomorphism the mapping pt(f ) : (pt(P 0 ), → (P 0 , P 1 , P 2 ) be an RSiCQuant-homomorphism.It is well known that the right adjoint map f * of f preserves prime elements.Hence the restriction pt(f ) of f * to pt(P ) gives the two maps First, we show that is strictly continuous.For this purpose we choose x ∈ Q 1 and p ∈ pt(P 0 ) with pt(f holds.Hence pt(f ) satisfies the condition (C 1 ) of the strict continuity which means that pt(f ) is strict continuous w.r.t. the noncommutative topologies O P 1 and O Q 1 .Since pt(Q 0 ) = A ⊤ and f is strong, i.e., f (⊤) = ⊤, then the axiom (C 3 ) of the strict continuity follows from Eq.( 2) and the following relation Hence the first map: is strictly continuous.Similarly, one can prove the strict continuity of the second map: and this prove the strict bicontinuity of the map: and this completes the proof.Now we introduce a functor and To study the adjunction between the functors: and the map: (1) The map is strictly bicontinuous, and (2) The system η = (η X ) X constitutes a natural transformation Id BiQS −→ P T • Ω. Proof.
(1) LetX = (X, O 1 (X), O 2 (X)) ∈ |BiQS| and x be an element of X.To prove the strict bicontinuity of we need to prove the strict continuity of both the mappings ) is strictly continuous.Similarly one can prove the strict continuity of the second map. ( ) be strictly bicontinuous.Since Ω(f ) is the left adjoint to pt(f ) we have which means that η = (η X ) X is a natural transformation from Id BiQS to P T • Ω.
there exists a unique strong homomorphism making the following diagram commutative: To prove the existence, let h(z) = f −1 (A z ).Obviously the mapping h : Q 0 −→ O 0 (X) is join preserving and fulfills the following property:

The same is true for the restrictions h |
Thus for G ∈ pt(O 0 (X)) we estimate from the right adjoint map h * (G) = {z ∈ Q : h(z) ≤ G} that the following relation holds: Uniqueness of h follows from the observation that given another RSiCQuanthomomorphism with the same property: Hence h is unique and it makes the diagram commutative.
is a quantale isomorphism, SpatCQuant will denote the full subcategory of the spatial coupled quantales in CQuant.Lemma 2.24.For all Q is a coupled quantale isomorphism.Lemma 2.25.For an (X, O 1 (X), O 2 (X)) ∈ |BiQS|, the idempotent and right-sided coupled quantales Proof.Clearly the map is a quantale isomorphism, which implies that the quantale O 1 (X) ∨ O 2 (X) is a spatial and therefore, the coupled quantales is spatial.
Lemma 2.27.For an Proof.Show bijectivity of the map η P T (Q) : P T (Q) → P T (Ω(P T (Q)) For injectivity, let p 1 , p 2 ∈ pt(Q 0 ) with p 1 = p 2 .Then there is a ∈ Q 0 with p 1 (a) = p 2 (a) i.e., there is O Q 0 (a) ∈ pt(Q 0 ) such that which shows that η P T (Q) (p 1 ) = η P T (Q) (p 2 ).Thus η P T (Q) is injective.To show the surjectivity of η P T (Q) , let q ∈ (pt(Q 0 ), O Q 1 , O Q 2 ) and put p = q • O Q 0 .Clearly p ∈ pt(Q 0 ).Furthermore, for all a ∈ Q 0 , we have which means that η P T (Q) is surjective.From this it follows that η P T (Q) is bijective, so the biquantum space (pt is sober, and the completes the proof.
From the above results, we have the following theorem.