FUZZY PROPERTIES IN FUZZY CONVERGENCE SPACES

Based on the concept of limit of prefilters and residual implication, several notions in fuzzy topology are fuzzyfied in the sense that, for each notion, the degree to which it is fulfilled is considered. We establish therefore theories of degrees of compactness and relative compactness, of closedness, and of continuity. The resulting theory generalizes the corresponding “crisp” theory in the realm of fuzzy convergence spaces and fuzzy topology.


Introduction.
In most papers and contributions to the theory of [0, 1]-topological spaces, the considered properties (like compactness) are viewed in a crisp way, that is, the properties either hold or fail.In [16], R. Lowen suggested that also the properties should be considered fuzzy, that is, one should be able to measure a degree to which a property holds.There are some papers dealing with such approaches.E. Lowen and R. Lowen [11] consider compactness degrees, and in [19], measures of separation in [0, 1]-topological spaces are investigated.In [17], Šostak developed a theory of compactness degrees and connectedness degrees in [0, 1]-fuzzy topological spaces, and developed, in [18], a theory of degrees of precompactness and completeness in the so-called Hutton fuzzy uniform spaces.These latter theories are related to the present work as they are explicitly based on a generalized inclusion (which is, however, not resulting from a residual implication).
In this paper, we follow these ideas in a systematic way.Starting from the notion of limit of a prefilter as defined in [15], we consider a semigroup operation * on [0, 1] which is finitely distributive over arbitrary joins and, therefore, has a right adjoint →, that is, a residual implication operator.In this way, a natural way of obtaining truly fuzzy extensions of properties in fuzzy convergence spaces [12,13] is to replace subsethood, a ≤ b of two fuzzy sets a, b ∈ [0, 1] X by degrees of subsethood, subset(a, b) = x∈X (a(x) → b(x)) [1].Exploiting this idea leads to the theory considered in this paper.We extend some results of an earlier paper [10], where degrees of closedness and degrees of compactness were studied and a theory of degrees of continuity and of degrees of relative compactness is established.Note that stratified [0, 1]-topological spaces [5,14] as well as Choquet convergence spaces [3] are fuzzy convergence spaces [13], that is, our approach works also in this more special context.

Preliminaries.
Throughout the paper, we consider fuzzy subsets with membership values in the real unit interval [0, 1], that is, a,b,c,... ∈ [0, 1] X .We assume that the reader is familiar with the usual definitions and notations in fuzzy set theory and fuzzy topology.We especially denote the pointwise extensions of the lattice operations ∧, ∨, and the order relation ≤ from [0, 1] to [0, 1] X again by ∧, ∨, ≤, respectively.Moreover, we write ∅ for the constant function 0. Further we want to consider an additional operation * : The proof is straightforward and therefore left to the reader.

Lemma 2.2. The residual implication has the following properties
Proof.Many of the assertions are easy consequences of (i) and can be found, for example, in [4].We only prove (ix (viii).From (i) the claim follows.

Fuzzy convergence spaces.
A prefilter (see [15]) We denote the set of all prefilters on a by F(a) and order this set by set inclusion.For F ∈ F(a) we denote by c(F) = f ∈F x∈X f (x) its characteristic value.A prefilter is called prime if whenever f ∨ g ∈ F then f ∈ F or g ∈ F [15].For example, the point prefilters [α1 x ] = {f ∈ F X (a) | f (x) ≥ α} are prime prefilters.We denote the set of all prime prefilters on a by F p (a).It is shown in [15] For further results concerning prefilters we refer the reader to [15].
For a fuzzy convergence space (a, lim) and b ≤ a, we denote b := b∈F∈Fp (a) limF its lim-closure [7].
If and call (b, lim| b ) a subspace of (a, lim) (cf.[6]). If For more details we refer to [6].

The degree of closedness of a fuzzy set.
The definitions and results of this section were already established in [10].However, we propose the proofs of the propositions in a more systematical way making use of Lemma 2.2.In [6], we called a fuzzy subset b ≤ a of a fuzzy convergence space (a, lim (ii) Put γ := λ∈Λ cl(b λ ).Then for every F ∈ F p (a) such that b λ ∈ F, for every x ∈ X, and for every λ ∈ Λ, we have From this the claim follows.
(iii) Its proof follows with condition (F1p) Section 3, as c(F) ≤ α for a∧α ∈ F ∈ F p (a) and by Lemma 2.2(ii).We end this section describing the degree of closedness for special operations * .For * = ∧ we obtain, with Lemma 2.1, and for * = T m , the Lukasiewicz t-norm, we deduce from Lemma 2.1 that is called the continuity degree of ϕ.
Obviously again it holds that ϕ is continuous if and only if cont(ϕ) = 1.

Degrees of compactness and of relative compactness.
In this section, we extend the theory of relative compact subsets established in [2,9] and repeat, sketching new proofs, the theory of compactness degrees developed in [10] (which extends the theory of compactness in fuzzy convergence spaces [6] and the theory of measures of compactness in [0, 1]-topological spaces [11]).Some additional results concerning compactness degrees are included.Clearly, (a, lim) is compact [6] if and only if c(a) = 1; and b is relatively compact in (a, lim) [9]   Proof.The proof of (i) was already shown in [10] and can be deduced similarly to (ii).We prove (ii).Corollary 6.11.The union of two compact (resp., relatively compact) fuzzy sets is compact (resp., relatively compact).
For the next proposition see also the related Proposition 3.4 in [10].
From this the claim follows.
Corollary 6.13.(i) The compactness degree of a lim-closed fuzzy subset of a fuzzy convergence space is at least as high as the compactness degree of the whole space.
(ii) A lim-closed fuzzy subset of a compact fuzzy convergence space is compact.
Corollary 6.15.(i) The compactness degree of the image of a fuzzy set under a continuous mapping is not smaller than the compactness degree of the fuzzy set.
(ii) The continuous image of a compact fuzzy set is compact.
(iii) The degree of relative compactness of the image of a fuzzy set under a continuous mapping is not smaller than the degree of relative compactness of the fuzzy set.
We conclude this section giving the compactness degrees and the degrees of relative compactness for special operations * .In case * = ∧ we get and in case * = T m we compute with Lemma 2.1 We mention without proof that in the case of a = 1 X , (X, ∆) a fuzzy topological space, the compactness degree for * = T m is just the degree of compactness in E. Lowen and R. Lowen [11].In this way, the compactness degrees here not only generalize the theory of compactness in FCS but also generalize the theory of compactness degrees in FTS, the category of fuzzy topological spaces.

7.
Conclusions.The theory of "truly" fuzzy properties developed in this paper relies mainly on the notion of residual implication with respect to the operation * .Hence it can easily be extended to more general situations, where the real unit interval is, for example, replaced by a more general lattice L. We have only to make sure, that the operation * : L × L → L then will still fulfill the properties (I), (II), (III), (IV), and (V) of Section 2 and that the residual implication → will fulfill the Lemma 2.2.For lattices that are suitable for this direction of research, we refer the reader to [5].

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 that the set P(F) := {G ∈ F p (a) | G ≥ F} contains minimal elements and we denote P m (F) := {G ∈ P(F) | G minimal}.We call B ⊂ F X (a) a prefilterbase if and only if ∅ ∉ B ≠ ∅ and b, c ∈ B ⇒ ∃d ∈ B, such that, d ≤ b ∧c.For a prefilterbase, we denote by [B] a = [B] = {f ∈ F X (a) | ∃b ∈ B : b ≤ f }, the generated prefilter.For b ≤ a and For F ∈ F(a), we define ϕ(F) as the prefilter on b generated by the prefilterbase {ϕ(f ) | f ∈ F}.If (a, lim a ), (b, lim b ) are fuzzy convergence spaces then we call ϕ : a → b continuous if and only if ϕ(lim a F) ≤ lim b ϕ(F) for all F ∈ F p (a).The category with fuzzy convergence spaces as objects and continuous mappings as morphisms is denoted by FCS.Let now (a, lim) ∈ |FCS| and let b ≤ a.We define on b the fuzzy convergence lim| b induced by (a, lim), lim| b F = lim[F] ∧ b, (3.1)

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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