The research of the axiomaticapproach as well as the constructive approach have also been extended to approximation operators in fuzzy environment. Morsi and Yakout [ 21 ] studied a set of axioms on fuzzyroughsetsbased on a triangular norm and a residual implicator, but their studies were restricted to the fuzzyrough set algebras constructed by fuzzy equivalence relations which were equivalence crisp relations when they degenerated into crisp ones. Thiele [ 22–24 ] investigated axiomatic characterizations of fuzzyrough approximation operators and roughfuzzy approximation operators within modal logic for fuzzy diamond and box operators. In [ 25 ], based on a fuzzy similarity relation, Radzikowska and Kerre define a broad family of the so called ( I , T ) -fuzzyroughsets which is determined by an implicator I and a triangular norm T . However, the properties and axiomatic characterization of ( I , T ) - fuzzyroughsets corresponding to an arbitrary fuzzy relation or a special fuzzy relation have not been studied. Wu et al. [ 26–28 ] examined many axioms on various classes of roughfuzzy and fuzzyrough approximation operators. Mi and Zhang [ 29 ] discussed axiomatic characterization of a pair of dual lower and upper fuzzy approximation operators based on a residual implication. Moreover, Liu [ 30 , 31 ] extended the axiomaticapproach to generalized roughsets over fuzzylattices, and Zhu [ 32 , 33 ] proposed the axiomatic system for covering basedrough set model. In [ 34 ], a further generalization of the notion of roughsets, called L-fuzzyroughsets, has been proposed by Radzikowska and Kerre. It differs from fuzzyroughsets extensively investigated in [ 22–29 , 35 , 36 ] in that it takes a complete residuated lattice L as its basic structure. This is a fairly wide constructive setting because diverse residuated pairs can be chosen and, in case L = [ 0 , 1 ] , the fuzzyrough set theory follows. However, the axiomatic characterization of L-fuzzyroughsets corresponding to an arbitrary L-fuzzy relation or a special L-fuzzy relation has not been studied, and the aim of the present paper is to investigate and to solve these questions.
In this paper, we built up a connection between roughsets, fuzzysets and lattices. Firstly, we introduced a new congruence relation induced by a fuzzy ideal of a distributive lattice, and then we presented a definition of lower and upper approximations of a subset of a distributive lattice with respect to a fuzzy ideal. Some properties of rough subsets in distributive lattices are investigated. Finally, we obtained that the notions of rough sublattices (ideals, filters), roughfuzzy sublattices (ideals, filters) are the extensions of sublattices (ideals, filters) and fuzzy sublattices (ideals, filters), respectively.
6 Conclusion and future work
In this paper, we have studied a general implicator-conjunctor based model for the lower and upper approximation of a fuzzy set under a binary fuzzy relation. We reviewed models from the literature that can be seen as special cases, and enriched the existing axiomaticapproach with a new notion of T -coupled pairs of approximations, which characterize the operations satisfying all relevant proper- ties of classical roughsets, i.e., left-continuous t-norms and their R-implicators. An important challenge is to extend the formal treatment to noise-tolerant fuzzyrough set models, such as those studied in [23–29]. Observing that the implicator-conjunctor based approximations are sensitive to small changes in the arguments (for instance, because of their reliance on inf and sup opera- tions), many authors have proposed models that are more robust against data perturbation. However, this normally goes at the expense of the properties the corresponding fuzzyrough set model satisfies.
Abstract. The aim of this paper is to introduce and study set- valued homomorphism on lattices and T -rough lattice with respect to a sublattice. This paper deals with T -rough set approach on the lattice theory. The result of this study contributes to, T -roughfuzzy set and approximation theory and proved in several papers. Keywords: approximation space; lattice; prime ideal; rough ideal; T -rough set; set-valued homomorphism; T -roughfuzzy ideal
Keywords: Roughsets; Fuzzysets; Fuzzyroughsets; Lower approximations; Upper approximations; Axioms
Rough set theory [16,17] is a new approach for reasoning about data. It has achieved a large amount of real applications such as medicine, information analysis, data mining, control and linguistics. As a tool to handling imperfect data, it complements other theories that deal with uncertain data, such as probability theory, evi- dence theory and fuzzy set theory. The main idea of roughsets corresponds to the lower and upper approx- imations. Pawlak’s deﬁnitions for the lower and upper approximations were originally introduced with reference to an equivalence relation. Many interesting properties of the lower and upper approximations have been derived by Pawlak [16,17] based on the equivalence relations. In this paper, we study a reverse problem. That is, can we characterize the notion of the lower and upper approximations in terms of those properties? We answer the question aﬃrmatively.
remainder of the process is the same. It also must be noted that |A| = |A 0 | R .
Along this work, the definition of cardinality for finite interval-valued hesitant fuzzysets has been tackled from an axiomatic point of view. Further- more, different results have been developed around this definition. From this perspective, cardinality is not fixed to a single function, but a wide range of mappings are considered as cardinalities.
A substantial experimental analysis of both the predictive ability and prediction stability of the lower and upper ap- proximation operators of four fuzzyrough set models has been carried out. This helps to verify how the predictions of these operators are influenced by increasing percentages of missing labels. We evaluated two different settings; one where only labelled instances are used in the predictions and a second where the unlabelled instances are also employed. The experimental evaluation shows that the first setting is generally preferred, although this may be due to implementing an approach for the second setting which is perhaps too naive. As future work, it would be interesting to verify whether more advanced heuristics may improve the performance of the sec- ond setting. One possible area of exploration revolves around the idea of allowing unlabelled instances to partially contribute to the construction of the fuzzyrough approximations.
Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved.
Fuzzy set theory was ﬁrst proposed by Zadeh  . It is an important mathematical approach to uncertain and vague data analysis, and has been widely used in the area of fuzzy decision making, fuzzy control, fuzzy inference, and so on [2–5] . Thereafter, the theory of rough set was proposed by Pawlak  , which was though of as another powerful tool for managing uncertainty that arises from inexact, noisy, or incomplete information. In terms of method, it was turned out to be method- ologically signiﬁcant in the domains of artiﬁcial intelligence and cognitive science, especially when representating or reason- ing with imprecise knowledge, machine learning and knowledge discovery. In recent years, the combination of fuzzy set theory and rough set theory has been studied by many researchers [7–13,31–35] , hence, many new mathematical methods are generated for dealing with the uncertain and imprecise information, such as the fuzzyroughsets and roughfuzzysets, etc. Meantime, many metric methods are presented and investigated by different authors, in order to measure the uncer- tainty and ambiguity of the different sets [14–20] .
All the above examples can be framed in turn into the general setting proposed here, and seen as particular instances of the general definition of L-fuzzy notion. But with this new general formulation, we do not restrict ourselves to the con- texts of IVF and AIF sets, and therefore we can cite additional examples from the literature that could not be included in our initial formulation proposed in , like the notions of inclusion and similarity between T2F sets introduced by Yang and Lin in , the notions of similarity and distance between T2F sets considered by Hung and Yang , different axiomatisations of similarity, distance and entropy between hesitant fuzzysets (see , , for instance) inclusion measures between hesitant fuzzysets , divergences between hesitant fuzzysets . Further examples in different generalised contexts can be found. Notwithstanding, an exhaustive list of the ax- iomatic definitions that can be regarded as particular instances of Definition 2 would fall outside the scope of this manuscript. One of the main features of this new formulation is that we do not need separate equations for every particular framework, but the same formula applies in different contexts, each of them referring to a particular instance of the lattice L. Thus, a single general L-fuzzy notion encompasses the original fuzzy notion, together with its possible extensions to more complex frameworks.
after that. When the dataset does not belong to any of the first three groups, the method decides to which of the final five it belongs, which are mutually exclusive. Having done so, the weight vector in (6) is set according to the weighting scheme advised for the selected group and the main algorithm devised by the user can be run. Examples of this approach are provided in Section 5.4.
In ranking based feature selection, the ﬁrst ‘‘k” best features are selected, where k is speciﬁed based on available domain knowledge. One can also add the best features one by one, and determine the classiﬁcation performance of the current fea- tures in each round until the classiﬁcation performance does not improve signiﬁcantly when adding more features. Here we compare the four evaluation measures when working with the second strategy. Datasets of iono, sonar, wdbc and wine are used in experiments. We employ linear SVM and RBF SVM to validate the selected features. Figs. 4–7 present the variation of classiﬁcation performance over the number of selected features. The results show that classiﬁcation accuracy increases with the number of selected features. The improvement is signiﬁcant at the beginning of the selection process. Afterwards, the classiﬁcation accuracy does not improve signiﬁcantly once a certain number of features have been selected. Considering the cost of classiﬁcation, we can delete the features which do not exhibit any signiﬁcant inﬂuence on the quality of classi- ﬁcation. Still we can ﬁnd that fuzzy entropy and dependency in Gaussian kernel approximation are competent with neigh- borhood roughsets and ReliefF. Entropy and dependency sometimes are better than the other two algorithms.
In this section we will deal with set-valued extensions of concepts originally defined within the context of fuzzysets. In , we have reviewed three different but related construction methods of set-valued generalizations, the so- called set-valued, max-min and max-min-varied extensions. Each of those construction methods includes, as particular cases, several particular definitions from the literature. Al- though less common, we can also find some works in the literature proposing lists of axioms that certain set-valued measures should satisfy. This is the case of the notion of set-valued inclusion measures of Cornelis-Kerre , and the three variants of the notion of interval-valued similarity measures respectively introduced by Bustince  Galar et al.  and Stachowiak-Dyczkowski . Let us recall Galar et al. definition  as an illustration of the general definition to be provided in this section (Definition 6):
Residuatedlattices and Stone algebras are two topics much studied in lattice theory, with impact on logic. One of the topics studied in residuatedlattices is the theory of the so-called implicative ideals and filters. This paper deals with algebras that are both residuatedlattices and Stone algebras, the subject-matter is the detection of properties of some special types of implicative filters (i-filters for short) that are specific to Stonean residuatedlattices. We mention that we introduce the notions of i-filter’s radical and boolean i-filter in Stonean ressiduated lattices. Several interesting results are obtained.
Commutative idempotent residuated Boolean algebras A residuated Boolean algebra or r-algebra (B 0 , ·, \, /) is a Boolean
algebra B 0 = (B, ∧, ∨, ¬, ⊥, ⊤) with three binary operations such that
xy ≤ z ⇐⇒ x ≤ z/y ⇐⇒ y ≤ x\z.
Abstract. In a real world, non-empty boundaries between classes may be both rough and fuzzy. In order to make decision in fuzzy approximation space, a fuzzy VPRS (variable precision rough set) approach is proposed based on substitution of the indiscernibility relation by a fuzzy indiscernibility relation in the rough approximation of decision classes, which can obtain probabilistic rules from fuzzy decision tables. Some set theoretic properties of the proposed approach are discussed.
Many generalizations of Pawlak’s rough set model can be found in literature. In , a survey on dual generalizations is presented. A pair of approximation operators is called dual, if for all A ⊆ U , apr(co(A)) = co(apr(A)).
Equivalence classes can be generalized by neighborhood operators. A neigh- borhood operator N is a mapping N : U → P(U ), where P(U ) represents the collection of subsets of U . It is often assumed that the neighborhood operator is reflexive, i.e., x ∈ N (x) for all x ∈ U . Given N , we define its inverse neighbor- hood operator N −1 by x ∈ N −1 (y) ⇔ y ∈ N (x) for x, y ∈ U .
it is isomorphic to a fuzzy opening system in some fuzzy powerset L X ;
(ii) every complete L-lattice is isomorphic to a fuzzy opening system in some fuzzy powerset L X if and only if ðL; ; 1Þ sat-
isﬁes the law of double negation.
Therefore, if ðL; ; 1Þ does not satisfy the law of double negation, then there exists a fuzzy complete lattice that is not iso- morphic to the concept lattice of any fuzzy context based on rough set theory. Thus, the expressive power of concept lattices of fuzzy contexts based on formal concept analysis is, in general, stronger than that based on rough set theory.
This section describes the survey related to the unimodal biometrics of palmprint used for identification. N.Duta et al. investigates the feasibility of person identification based on feature points extracted from palmprint images. They proposed following three paradigms used for palm matching. This set of feature points are extracted from palm lines; pairwise distance is computed for feature points and verifies the identity of palmprints. They also mention some limitations of palm print matching for further research. N. Duta  gives a survey on hand shape based biometric systems. The survey consists of review of component modules including the employed algorithms, system taxonomies, performance evaluation methodologies, summary of the accuracy results reported in the literature, testing issues, commercial hand shape biometric systems and evaluations. They have also mentioned few limitations of the hand shape biometric and provided some directions for future research. D. Zhang et al., presented online palm print identification using CCD cameras. A constant palmprint image is acquired by using a case and a cover. Due to the fixed background and uniform illumination, the palm images are segmented using otus method. They have proposed 2D Gabor phase for feature extraction and classified using tangent and bisector based techniques. They gave some future directions to reduce the size of the device and some other feature extraction code and different classifier to use for higher performance. C. L. Lin et al. gives reliable and robust personal verification approach using palmprint features. Palm features does not require about objects and the parameters. Two finger-webs are automatically selected as datum points for region of interest (ROI) in palm images
Let us recall the necessary background. Let K be an arbitrary variety of commutative, integral residuatedlattices. The paper  studies two finitary logics associated with each such K, which are denoted by their consequence relations: The first one, denoted by K , is the truth-preserving logic determined by the algebras in K when their maximum 1 (which is also the unit of the monoid structure of the fusion operation ) is taken as representing truth; the second one, denoted by |= K , is the logic
In this paper, we introduce the concepts of upper and lower rough hyper fuzzy ideals (filters) in a hyperlattice and their basic properties are discussed. Let θ be a hyper congruence relation on L. We show that if µ is a fuzzy subset of L, then θ(< µ >) = θ(< θ(µ) >) and θ(µ ∗ ) = θ((θ(µ)) ∗ ), where < µ > is the least hyper fuzzy ideal of L containing µ and