TRANSITIVITY IN UNIFORM APPROACH THEORY

We introduce a notion of transitivity for approach uniformities and approach uniform convergence spaces, yielding reflective subconstructs of AUnif 
and AUCS. Further, we investigate how these new categories are related to uACHY, uACHY U , and uMET, and we show that these relationships are similar to those in the classical case.


Introduction.
Since the first considerations on zero-dimensional spaces, by F. Hausdorff, and the original study of non-Archimedean metric spaces, by A. F. Monna, the amount of literature on transitive structures has become extensive.Transitivity turned out to be interesting in a wide range of fields (functional analysis, Boolean algebra, valuation rings, domain theory, and many others) which proves the great importance of the concept.Therefore, an investigation of this topic in the setting of uniform approach structures is inevitable.
This paper presents a transitivity condition for two important quantified uniform structures: one for approach uniformities (introduced in Lowen and Windels [5] as a quantification of Unif) and a related concept for approach uniform convergence spaces (introduced in Windels [7] as a quantification of UCS).These definitions in turn yield different transitivity concepts in the setting of approach Cauchy spaces (introduced in Lowen and Lee [4] as a quantification of CHY).
Although the categories UCS and CHY are well known to be Cartesian closed (see Lee [3] and Bentley et al. [1], respectively), the associated quantified structures yield categories which do not share this property; the triangle inequality-like axiom turned out to be the essential problem.One possible solution, which is discussed in [7], is to drop this particular axiom.Alternatively, we can demand a stronger (non-Archimedean) triangle inequality to be fulfilled: in the case of Cauchy spaces, this approach leads to the Cartesian closed category uACHY (see [4]).In this paper, we will pursue the same method for uniform convergence spaces.
For any set X, we denote the set of all filters on X by Ᏺ(X).The filter generated by a filter basis Ꮾ is denoted by [Ꮾ].In particular, the point filter generated by the set {x} is denoted by ẋ.If Ᏺ, Ᏻ ∈ Ᏺ(X), then provided that every U • V = {(x, y) ∈ X × X : there exists z ∈ X such that (x, z) ∈ U and (z, y) ∈ V } is not empty; whenever this notation is used, we will tacitly assume this condition to be fulfilled.
Recall from [8] that a semi-uniform convergence structure L on a set X is a collection of filters on X × X such that (UCS1) ẋ × ẋ ∈ L for all x ∈ X, (UCS2) if Φ ∈ L and Φ ⊂ Ψ , then The collection L is called a uniform convergence structure if it also satisfies the supplementary condition The pair (X, L) is called a uniform convergence space.
For any semi-uniform convergence spaces (X, L) and (Y , K), a map f : Let UCS denote the category of uniform convergence spaces and uniformly continuous maps.

The category uAUCS.
In this section, we introduce a notion of transitivity for approach uniform convergence structures.Recall from Windels [7] that an approach uniform convergence structure on a set X is a map η : Ᏺ(X ×X) → [0, ∞] satisfying the following conditions: for all x ∈ X and all Φ, Ψ ∈ Ᏺ(X × X), Alternatively, such a structure can be described by a uniform convergence tower The equivalence is shown by considering The pair (X, η) (or, equivalently, the pair (X, (L ε ) ε∈R + )) is called an approach uniform convergence space (AUC-space for short).
Given AUC-spaces (X, η) and (Y , η ) with uniform convergence towers (L ε ) ε and Let AUCS denote the category of AUC-spaces and uniform contractions.For details, the reader is referred to [7].
Definition 2.1.Let X be a set.An AUC-structure η : Ᏺ(X × X) → [0, ∞] is called an ultra approach uniform convergence structure if it satisfies instead of (AUCS5) the stronger condition: The pair (X, η) is called an ultra approach uniform convergence space (uAUC-space for short).uAUC-spaces can be described by uniform convergence towers too.Proposition 2.2.Let (X, η) be an AUC-space, and let (L ε ) ε∈R + denote its uniform convergence tower.Then, the following are equivalent: (1) (X, η) is an ultra approach uniform convergence space, (2) for every ε ∈ R + , L ε is a uniform convergence structure.
Let uAUCS denote the full subcategory of AUCS consisting of all uAUC-spaces.
Proof.For a family ((X j ,η j )) j∈J of uAUC-spaces and a source (X f j →(X j ,η j )) j∈J in AUCS, the initial approach uniform convergence structure η : satisfies (uAUCS5).For this, let Φ, Ψ ∈ Ᏺ(X × X) be such that there exists Φ • Ψ , then for each j ∈ J, (f j × f j )(Φ) • (f j × f j )(Ψ ) exists and So uAUCS is initially closed in AUCS and since uAUCS contains all indiscrete objects, this proves the claim.
Proof.This is an immediate consequence of Theorem 2.3 and [2, Theorem A.10].
Initial sources can be described by means of towers as well.
Proposition 2.5.Let (X, η) and ((X j ,η j )) j∈J be uAUC-spaces, and let (L ε ) ε and (L j ε ) ε denote the respective towers.Then, the following are equivalent: (1) ((X, η) Proof.For every ε ∈ R + , let K ε be the initial uniform convergence structure for the source (X and thus we have which proves the claim. For any uAUC-spaces (X, η) and (Y , η ), let C(X, Y ) be the set of all uniform contractions from X to Y .Then, for any Φ ∈ Ᏺ(X × X) and where (2.5) Proposition 2.6.The map η * yields the coarsest uAUC-structure on C(X, Y ) with respect to which the evaluation map ev : Proof.Clearly, η * is well defined.(AUCS1) follows from the inequality and the converse follows from (AUCS2).Since for any )) be such that there exist Θ • Θ and Φ ∈ Ᏺ(X × X).Then for any H ∈ Θ, K ∈ Θ , and A ∈ Φ, it holds that where π 1 and π 2 are the canonical projection maps from X ×C(X, Y ) to X and C(X, Y ), respectively, the map ev : X ×C(X, Y ) → Y is a uniform contraction with respect to η * .Let η * be another uAUC-structure on C(X, Y ) with respect to which ev is a uniform contraction.Then for all Φ ∈ Ᏺ(X × X) and ) and hence we have the result.
Proposition 2.7.Let (X, η), (Y , η ), and (Z, η ) be uAUC-spaces and let f : X ×Z → Y be a uniform contraction.Then there exists a unique uniform contraction f : (2.11) x z for all x ∈ X.Since the identity map, the constant map, and the composition of uniform contractions are uniform contractions, f is a uniform contraction and hence the map f is well defined.Furthermore, for any Φ ∈ Ᏺ(X × X) and Ψ ∈ Ᏺ(Z × Z), we have (2.12) Combining Propositions 2.6 and 2.7, we have the following theorem.
For any uniform convergence space (X, L), the map η is clearly an uAUC-structure on X.Furthermore, for any uniform convergence spaces So UCS is embedded as a full subcategory in uAUCS by the functor and analogously to [7, Proposition 11], we have the following proposition.
Theorem 2.10.The category UCS is a bicoreflective subcategory of uAUCS.
Theorem 2.11.The category UCS is a bireflective subcategory of uAUCS.
3. The category AUnif U .In this section and in Section 4 we discuss two different notions of transitivity for approach uniformities.Recall from Lowen and Windels [5] that an approach uniformity on a set X, is an ideal ᐁ of functions from X × X into [0, ∞], satisfying the following conditions: (AU1) for all u ∈ ᐁ, for all x ∈ X : Equivalently, an approach uniformity can be described with a uniform tower, that is, a family of semi-uniformities The equivalence is shown by considering ᐁ ε = {{u < α} : α > ε, u ∈ ᐁ}.The pair (X, ᐁ) (or, equivalently, the pair (X, (ᐁ ε ) ε∈R + )) is called an approach uniform space.
The function f : The category of approach uniform spaces and uniform contractions is denoted by AUnif.For details, the reader is referred to [5].Definition 3.1.An approach uniform space (X, (ᐁ ε ) ε∈R + ) satisfying the supplementary condition that every ᐁ ε is a uniformity, is called level-uniform.
Proof.Let ((X, η) f j →(X j ,η j )) j∈J be an initial source in uAUCS, and suppose that every (X j ,η j ) is level-uniform.If (L ε ) ε is the tower of η and for all j ∈ J : (L j ε ) ε is the tower of η j , then, by Proposition 2.5, for all ε ∈ R + : ((X, L ε ) and since Unif is a reflective subcategory of UCS, every (X, L ε ) is level-uniform.Consequently, (X, η) is level-uniform.Thus AUnif U is initially closed in AUCS.Furthermore, since AUnif U contains all indiscrete objects, we have the result.The category AUnif U is not Cartesian closed, since it contains Unif both reflectively and coreflectively.

The category tAUnif.
Since AUnif contains both the category of uniform spaces and the category of pseudo-metric spaces, it is natural to seek a subcategory of AUnif that generalizes the notions of transitive uniform spaces and ultra-metric spaces.Recall that a uniform space (X, ᐁ) is called transitive if ᐁ has a basis of entourages U with the property that U •U = U .A pseudo-metric d on X is called an ultra-pseudometric (or non-Archimedean pseudo-metric) if d satisfies the strong triangle inequality d(x, z) ≤ d(x, y) ∨ d(y, z) for every x, y, z ∈ X.
Every approach uniformity induced by a transitive uniformity or by an ultra-metric is level-uniform, but not vice versa.In fact, every uniformly generated approach uniformity is level-uniform.This section establishes a stronger notion of transitivity for approach uniformities, in order to eliminate this disadvantage.Since every approach uniformity has a basis of pseudo-metrics, it seems natural to adopt the following definition.Definition 4.1.An approach uniform space (X, ᐁ) is called transitive if ᐁ has a basis consisting of ultra-pseudo-metrics.
Let tAUnif denote the full subcategory of AUnif consisting of all transitive approach uniformities.
Theorem 4.3.The category tAUnif is a reflective subcategory of AUnif U .Consequently, tAUnif is a topological construct.
Proof.Since AUnif U is a reflective subcategory of AUCS, initial structures in both categories are the same.Therefore the same argument as for Theorems 3.3 and Proposition 3.4 can be used.
Proof.By virtue of Proposition 4.2, this is evident.
Proof.To see that (1)⇒(2), suppose that Ꮾ is a basis for ᐁ consisting of ultrapseudo-metrics.Then d = sup u∈Ꮾ u, and therefore d is an ultra-pseudo-metric too.The converse is trivial, since {d} is a basis for ᐁ.
The categories of ultra-pseudo-metric spaces and of transitive uniform spaces are nicely embedded in tAUnif, analogously to the classical case.The category tAUnif is not Cartesian closed, since it contains tUnif both reflectively and coreflectively, and tUnif is not Cartesian closed (in fact, any reflective subcategory of Unif containing a nondiscrete object is not Cartesian closed).

Embedding uACHY in uAUCS.
Recall from Lee and Lowen [4] that a function γ : Ᏺ(X) → [0, ∞] is called an ultra approach Cauchy structure (for short, uACHYstructure) on X if it satisfies the following conditions: The pair (X, γ) is called an ultra approach Cauchy space (for short, uACHY-space).
Proof.(1) For any Ᏺ ∈ Ᏺ(X), take any (Ᏺ j ) n j=1 ∈ β(Ᏺ × Ᏺ).Without loss of generality, we may assume Ᏺ i ∨ Ᏺ j does not exist for i = j and hence we can take (A j ) n j=1 such that A j ∈ Ᏺ j for each j = 1,...,n and A i ∩ A j = ∅ for i = j.Then there exists F ∈ Ᏺ such that F × F ⊆ n j=1 (A j × A j ) and so F ⊆ A k for some k ∈ {1,...,n}.Since A j ∩ A k = ∅ for j = k, we get Ᏺ k ⊂ Ᏺ and consequently γ(Ᏺ k ) ≥ γ(Ᏺ).Thus γ ≤ γ ηγ and the converse is obvious.
6.The categories uACHY U and uACHY tU .Throughout this section (X, ᐁ) will be a level-uniform approach uniform space, and (ᐁ ε ) ε will denote its uniform tower.Then the map γ is an ultra approach Cauchy structure on X. Conditions (AF1) and (AF2) are obvious and (uACHY) is immediate from (UT4) and the fact that each ᐁ ε is a uniform structure on X.We say that γ ᐁ and ᐁ are compatible and γ ᐁ is called the uACHY-structure induced by ᐁ.Given a set X, an uACHY-structure γ on X is said to be approach uniformizable if there exists a compatible AUnif U -structure ᐁ on X, that is, γ = γ ᐁ for some AUnif U -structure ᐁ on X.
Let uACHY U be the full subcategory of uACHY consisting of all approach uniformizable uACHY-spaces (for short, uACHY U -spaces).
Proof.For any family ((X j ,γ j )) j∈J of uACHY U -spaces, say γ j = γ ᐁ j , and any source X f j → X j ,γ j j∈J (6.6) in uACHY, let γ be the initial uACHY-structure on X and let ᐁ be the initial AUnif Uuniform tower on X for the source Then ᐁ induces γ by Proposition 6.1 and hence uACHY U is initially closed in uACHY.Furthermore, since uACHY U contains all indiscrete objects, we have the result.
Proof.For any uACHY U -space (X, γ), let C(X, γ) be the collection of all contractions from (X, γ) to uACHY U -spaces and let ᐁ be the initial AUnif U -structure on X for the source X g → Z, ᐁ γ g∈C(X,γ) (6.8) in AUnif U .Then γ ᐁ is the initial uACHY U -structure on X for the source So by the fact that the identity map is a contraction, we get 1 Hence there is a functor Proof.For any uACHY U -structure γ on a set X, we have γ ᐁγ = γ and for any AUnif U -structure ᐁ on X, ᐁ γ ᐁ is finer than ᐁ.So by [6, Theorem 2.2.10], for any AUnif U -space (X, ᐁ) is the uACHY U -bicoreflection.
For any set X, a uACHY U -structure γ on X is said to be transitively approach uniformizable if γ is compatible with some transitive approach uniformity ᐁ on X.
Let uACHY tU be the full subcategory of uACHY consisting of all transitive approach uniformizable uACHY-spaces (for short, uACHY tU -spaces).
Since tAUnif is initially closed in AUnif U , we have the following theorem.
Theorem 6.5.The category uACHY tU is a bireflective subcategory of uACHY U .
For any uACHY tU -space (X, γ), let ᐁ t γ be the initial transitive approach uniformity with respect to the source (X , where ᐁ t (γ) is the class of all transitive approach uniformities inducing γ.Then ᐁ t γ is the finest transitive approach uniformity on X inducing γ.Clearly the restriction of the above two functors are well defined and we get the analogous result.The categories tUnif and CHY tU are the full subcategories of Unif and CHY U whose objects are transitive uniform spaces and transitive uniformizable Cauchy spaces, respectively.These categories form a similar diagram as Unif and CHY U .At the end of Section 2, we showed that UCS is both reflectively and coreflectively embedded in uAUCS.The argument is representative for all upward arrows in the diagram.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning where H(A) = {(h(a), k(b)) : (a, b) ∈ A, (h, k) ∈ H} for each A ∈ Φ and H ∈ Θ, forms a filter basis on Y × Y .Let Θ(Φ) be the filter on Y × Y generated by this basis and define a map η *

Proposition 3 . 4 .
The category AUnif U is a topological construct.Proof.This is an immediate consequence of Theorem 3.3 and [2, Theorem A.10].

Theorem 4 . 6 .
The category tUnif is a bireflective and bicoreflective subcategory of tAUnif.The category uMET is a bicoreflective subcategory of tAUnif.Therefore, we have the following diagram:

Theorem 6 . 6 . 7 .
The category uACHY tU is a bicoreflective subcategory of tAUnif.Categorical overview.Summarizing the results in foregoing sections, we obtain the following diagram:

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation