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Rough Set (RS) is a relatively new technique for making imprecise and uncertain decisions. It is an emerging area of ambiguous mathematics associated with fuzzy set theory. In comparison with fuzzy sets and other theories, RS has more advantages. Unlike statistics, evidence theory, and fuzzy set theory, RS theory does not require any additional or preliminary information about data. RS theory deals with approximating arbitrary subsets of the universe using two subsets namely lower and upper approximation. The advantage of RS is that rough approximations can be expanded into fuzzy environments which aids in solving a wide variety of realtime problems. The goal of this study is to focus on Rough Fuzzy Prime Ideals (RFPI) in ring structure and explain specific aspects of its upper and lower approximations. Furthermore, Rough Fuzzy Ideal (RFI) and RFPI notions and characteristics are described. [ABSTRACT FROM AUTHOR]

The relationship between Rough Set (RS) and algebraic systems has been long studied by mathematicians. RS is a growing research area that encourages studies into both real-world applications and the theory itself. In RS, a universe subset is characterized by a pair of ordinary sets called lower and upper approximations. In this study, we look attentively at the use of rough sets when the universe set has a ring structure. The main contribution of the paper is to concentrate on the study of rough fuzzy ideals concerning the gamma ring and to describe some properties of its lower and upper approximations. This paper deals with the connection between Rough Fuzzy Sets (RFS) and ring theory. The goal of this paper is to present the notion of Left Operator Rings (LOR) and Right Operator Rings (ROR) in the gamma ring structure. We introduce some basic concepts of rough fuzzy left and right operator rings. Furthermore, we investigate some characterizations of left and right operator rings and prove some theorems based on these results. [ABSTRACT FROM AUTHOR]

Since Pawlak defined the notion of rough sets in 1982, many authors made wide research studying rough sets in the ordinary case and the fuzzy case. This paper introduced a new style of rough fuzzy sets based on a fuzzy ideal ℓ on a universal finite set X. New lower and new upper fuzzy sets are introduced, and consequently, fuzzy interior and fuzzy closure operators of a rough fuzzy set are discussed. These definitions, if ℓ is restricted to ℓ° = {ō}, imply the fuzzification of previous definitions given in the ordinary case, and moreover in the crisp case, we get exactly these previous definitions. The new style gives us a better accuracy value of roughness than the previous styles. Rough fuzzy connectedness is introduced as a sample of applications on the recent style of roughness. [ABSTRACT FROM AUTHOR]

In this paper we expose interval rough fuzzy ideals in γ-near-rings. Also we prove some interesting properties of γ-ideal and bi-γ-ideal in γ-nearring. [ABSTRACT FROM AUTHOR]

In the philosophy of rough set theory, the methodologies of rough soft sets and rough fuzzy sets have been being examined to be efficient mathematical tools to deal with unpredictability. The basic of approximations in rough set theory is based on equivalence relations. In the aftermath, such theory is extended by arbitrary binary relations and fuzzy relations for more wide approximation spaces. In recent years, the notion of picture hesitant fuzzy relations by Mathew et al. can be considered as a novel extension of fuzzy relations. Then this paper proposes extended approximations into rough soft sets and rough fuzzy sets from the viewpoint of its. We give corresponding examples to illustrate the correctness of such approximations. The relationships between the set-valued picture hesitant fuzzy relations with the upper (resp., lower) rough approximations of soft sets and fuzzy sets are investigated. Especially, it is shown that every non-rough soft set and non-rough fuzzy set can be induced by set-valued picture hesitant fuzzy reflexive relations and set-valued picture hesitant fuzzy antisymmetric relations. By processing the approximations and advantages in the new existing tools, some terms and products have been applied to semigroups. Then, we provide attractive results of upper (resp., lower) rough approximations of prime idealistic soft semigroups over semigroups and fuzzy prime ideals of semigroups induced by set-valued picture hesitant fuzzy relations on semigroups.

It is well agreed that geologic risk occurs during hydrocarbon exploration because diverse uncertainties accompany the entire hydrocarbon system parameters such as the source rock, reservoir rock, trap and seal rock. In order to overcome such attributes with uncertainties, a number of soft computing methods are used. Information granules could be provided by the Rough Fuzzy Set Granulation (RFSG) with a thorough quality evaluation. This is capable of attribute reduction that has been claimed to be essential in investigating the hydrocarbon systems. This paper is an endeavor to recommend a Geospatial Information System (GIS)-based model with the aim of categorizing the hydrocarbon structures map consistent with the uncertainty range concepts of geologic risk in the rough fuzzy sets and granular computing. The model used the RFSG for the attribute reduction by a Decision Logic language (DL-language). The RFSG was employed in order to classify hydrocarbon structures according to geological risk and extract the fuzzy rules with a predefined range of uncertainty. In order to assess the precisions of the fuzzy decisions on the hydrocarbon structure classification, the fuzzy entropy and fuzzy cross-entropy are applied. The proposed RFSG model applied for 62 structures as the training data, average fuzzy entropy has been calculated as 0.85, whereas the average fuzzy cross-entropy has been calculated 0.18. As it can be discerned, just seven structures had cross-entropies greater than 0.1, while three structures were larger than 0.3. It is implied that the precision of the proposed model is about 89%. The results yielded two reductions for the condition attributes and 11 fuzzy rules being filtered by the granular computing values.

The relationship between Rough Set (RS) and algebraic systems has been long studied by mathematicians. RS is a growing research area that encourages studies into both real-world applications and the theory itself. In RS, a universe subset is characterized by a pair of ordinary sets called lower and upper approximations. In this study, we look attentively at the use of rough sets when the universe set has a ring structure. The main contribution of the paper is to concentrate on the study of rough fuzzy ideals concerning the gamma ring and to describe some properties of its lower and upper approximations. This paper deals with the connection between Rough Fuzzy Sets (RFS) and ring theory. The goal of this paper is to present the notion of Left Operator Rings (LOR) and Right Operator Rings (ROR) in the gamma ring structure. We introduce some basic concepts of rough fuzzy left and right operator rings. Furthermore, we investigate some characterizations of left and right operator rings and prove some theorems based on these results.

It is well agreed that geologic risk occurs during hydrocarbon exploration because diverse uncertainties accompany the entire hydrocarbon system parameters such as the source rock, reservoir rock, trap and seal rock. In order to overcome such attributes with uncertainties, a number of soft computing methods are used. Information granules could be provided by the Rough Fuzzy Set Granulation (RFSG) with a thorough quality evaluation. This is capable of attribute reduction that has been claimed to be essential in investigating the hydrocarbon systems. This paper is an endeavor to recommend a Geospatial Information System (GIS)-based model with the aim of categorizing the hydrocarbon structures map consistent with the uncertainty range concepts of geologic risk in the rough fuzzy sets and granular computing. The model used the RFSG for the attribute reduction by a Decision Logic language (DL-language). The RFSG was employed in order to classify hydrocarbon structures according to geological risk and extract the fuzzy rules with a predefined range of uncertainty. In order to assess the precisions of the fuzzy decisions on the hydrocarbon structure classification, the fuzzy entropy and fuzzy cross-entropy are applied. The proposed RFSG model applied for 62 structures as the training data, average fuzzy entropy has been calculated as 0.85, whereas the average fuzzy cross-entropy has been calculated 0.18. As it can be discerned, just seven structures had cross-entropies greater than 0.1, while three structures were larger than 0.3. It is implied that the precision of the proposed model is about 89%. The results yielded two reductions for the condition attributes and 11 fuzzy rules being filtered by the granular computing values. [ABSTRACT FROM AUTHOR]

Fuzzy sets are a major simplification and wing of classical sets. The extended concept of set theory is rough set (RS) theory. It is a formalistic theory based upon a foundational study of the logical features of the fundamental system. The RS theory provides a new mathematical method for insufficient understanding. It enables the creation of sets of verdict rules from data in a presentable manner. An RS boundary area can be created using the algebraic operators union and intersection, which is known as an approximation. In terms of data uncertainty, fuzzy set theory and RS theory are both applicable. In general, as a uniting theme that unites diverse areas of modern arithmetic, symmetry is immensely important and helpful. The goal of this article is to present the notion of rough fuzzy ideals (RFI) in the gamma ring structure. We introduce the basic concept of RFI, and the theorems are proven for their characteristic function. After that, we explain the operations on RFI, and related theorems are given. Additionally, we prove some theorems on rough fuzzy prime ideals. Furthermore, using the concept of rough gamma endomorphism, we propose some theorems on the morphism properties of RFI in the gamma ring. [ABSTRACT FROM AUTHOR]

This paper expose a study on rough interval valued pythagorean fuzzy sets in semigroups. We characterize rough interval valued pythagorean fuzzy sets by an exam- ple. Characterize composition of two interval valued pythagorean fuzzy sets. Introduce rough interval valued pythagorean fuzzy left(right, bi-, interior-,(1,2)-)ideals in semi- groups. Moreover we prove an interval valued pythagorean fuzzy set is an upper rough interval valued pythagorean fuzzy left(right) ideal of semigroup also we give an example for converse of this is not true. Lower and upper approximation of an interval valued pythagorean fuzzy ideal of semigroup is an interval valued pythagorean fuzzy ideal of semigroup. [ABSTRACT FROM AUTHOR]

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Fuzzy sets are a major simplification and wing of classical sets. The extended concept of set theory is rough set (RS) theory. It is a formalistic theory based upon a foundational study of the logical features of the fundamental system. The RS theory provides a new mathematical method for insufficient understanding. It enables the creation of sets of verdict rules from data in a presentable manner. An RS boundary area can be created using the algebraic operators union and intersection, which is known as an approximation. In terms of data uncertainty, fuzzy set theory and RS theory are both applicable. In general, as a uniting theme that unites diverse areas of modern arithmetic, symmetry is immensely important and helpful. The goal of this article is to present the notion of rough fuzzy ideals (RFI) in the gamma ring structure. We introduce the basic concept of RFI, and the theorems are proven for their characteristic function. After that, we explain the operations on RFI, and related theorems are given. Additionally, we prove some theorems on rough fuzzy prime ideals. Furthermore, using the concept of rough gamma endomorphism, we propose some theorems on the morphism properties of RFI in the gamma ring.

Multigranulation rough set is one class of the important models in rough set community. However, both pessimistic and optimistic rough sets have disadvantages in describing target concept. In this paper, a novel model, called weighted multigranulation fuzzy decision rough sets, is proposed. Gaussian kernel is used to compute the similarity between objects, which induces a fuzzy equivalence relation. We employ the relation to fuzzily partition the universe and then obtain multiple fuzzy granulations from multisource fuzzy information system. Moreover, some interesting properties of the proposed model are discussed. A comparative study between the proposed multigranulation model and Sun's multigranulation rough set model is carried out. An example is employed to illustrate the effectiveness of the proposed method, which may provide an effective approach for multisource data analysis in real applications. [ABSTRACT FROM AUTHOR]

Three-way decisions model proposed by Yao gives a semantic interpretation of positive region, negative region and boundary region. This model was developed in the framework of classical rough set, the approached target concept is a crisp set, the employed knowledge is a equivalence relation. In this paper, we extend the three-way decisions model to rough fuzzy set. Specifically, the target concept is extended to a fuzzy set, while the used knowledge is also a equivalence relation. An example is given to illustrate the computation processes of the proposed three-way decisions model. The extended model can deal with the problems described by fuzzy decision tables with symbolic-valued conditional attributes and fuzzy decision attributes. [ABSTRACT FROM AUTHOR]

The original Pawlak's rough set approach based on indiscernibility relation (single granularity) has been extended to multigranulation rough set structure in the recent years. Multigranulation rough set approach has become a flouring research direction in rough set theory. This paper considers rough approximation of a fuzzy concept under the framework of multigranulation over two different universes of discourse, i.e., multigranulation fuzzy rough set models over two universes. We present three types of multigranulation fuzzy rough set over two universes by the constructive approach, respectively. Some interesting properties of the proposed models are discussed and also the interrelationships between the proposed models and the existing rough set models are given. We then propose a new approach to a kind of multiple criteria group decision making problem based on multigranulation fuzzy rough set model over two universes. The decision rules and algorithm of the proposed method are given and an example of handling multiple criteria group decision making problem of clothes ranking illustrates this approach. The main contribution of this paper is twofold. One is to establish the multigranulation fuzzy rough set theory over two universes. Another is to try presenting a new approach to multiple criteria group decision making based on multigranulation fuzzy rough set over two universes. The proposed models not only enrich the theory of multigranulation rough set but also make a tentative to provide a new perspective for multiple criteria group decision making with uncertainty. [ABSTRACT FROM AUTHOR]

In a dynamic environment, the data collected from real applications varies not only with the amount of objects but also with the number of features, which will result in continuous change of knowledge over time. The static methods of updating knowledge need to recompute from scratch when new data are added every time. This makes it potentially very time-consuming to update knowledge, especially as the dataset grows dramatically. Calculation of approximations is one of main mining tasks in rough set theory, like frequent pattern mining in association rules. Considering the fuzzy descriptions of decision states in the universe under fuzzy environment, this paper aims to provide an efficient approach for computing rough approximations of fuzzy concepts in dynamic fuzzy decision systems (FDS) with simultaneous variation of objects and features. We firstly present a matrix-based representation of rough fuzzy approximations by a Boolean matrix associated with a matrix operator in FDS. While adding the objects and features concurrently, incremental mechanisms for updating rough fuzzy approximations are introduced, and the corresponding matrix-based dynamic algorithm is developed. Unlike the static method of computing approximations by updating the whole relation matrix, our new approach partitions it into sub-matrices and updates each sub-matrix locally by utilizing the previous matrix information and the interactive information of each sub-matrix to avoid unnecessary calculations. Experimental results on six UCI datasets shown that the proposed dynamic algorithm achieves significantly higher efficiency than the static algorithm and the combination of two reference incremental algorithms. [ABSTRACT FROM AUTHOR]

The extension of rough set model is an important research direction in rough set theory. This paper presents two new extensions of the rough set model over two different universes. By means of a binary relation between two universes of discourse, two pairs of rough fuzzy approximation operators are proposed. These models guarantee that the approximating sets and the approximated sets are on the same universes of discourse. Furthermore, some interesting properties are investigated, the connections between relations and rough fuzzy approximation operators are examined. Finally, the connections of these approximation operators are made, and conditions under which these approximation operators made equivalent are obtained. [ABSTRACT FROM AUTHOR]

Based on Bayesian decision theory, three-way decisions model (TWDM) proposed by Yao gives new semantic interpretations of positive region, negative region and boundary region. Some extensions of TWDM have been proposed by different authors and have been successfully applied to many fields, such as soft computing, data mining and decision making. However existing three-way decisions models are almost developed in certainty environment, which limits their applications in uncertainty environment. In order to deal with this problem, based on tolerance rough fuzzy set, a TWDM is proposed in this paper. The main contributions of this paper include two aspects: (1) the tolerance rough fuzzy set which is extended from rough fuzzy set is introduced, and some basic properties of the tolerance rough fuzzy set are investigated. (2) The TWDM with respect to the tolerance rough fuzzy set is proposed. In addition, an example is given to illustrate the computation processes of the TWDM. [ABSTRACT FROM AUTHOR]

In rough fuzzy set, equivalence relation is used for approximating the target concept, which is a fuzzy set. In this paper, we extend the equivalence relation in rough fuzzy set to tolerance relation, and propose an extended model named tolerance rough fuzzy set. Furthermore, the properties of the extended model are investigated, and the proofs of the properties are given in the paper. The advantage of the extended model is that it can directly deal with continuous-valued attributes, and it does not require the process of discretization. [ABSTRACT FROM AUTHOR]

Considering the incompleteness and uncertainty of information systems, this paper combines the incomplete information systems with interval-valued fuzzy sets, and defines the incomplete interval-valued fuzzy information systems. The basic rough set theory for incomplete interval-valued fuzzy information systems is also discussed in this paper. [ABSTRACT FROM AUTHOR]

Abstract: With the development of data mining and soft computing techniques, it becomes possible to automatically mine knowledge from spatial data. Spatial rule extraction from spatial data with uncertainty is an important issue in spatial data mining. Rough set theory is an effective tool for rule extraction from data with roughness. In our previous studies, Rough set method has been successfully used in the analysis of social and environmental causes of neural tube birth defects. However, both roughness and fuzziness may co-exist in spatial data because of the complexity of the object and the subjective limitation of human knowledge. The situation of fuzzy decisions, which is often encountered in spatial data, is beyond the capability of classical rough set theory. This paper presents a model based on rough fuzzy sets to extract spatial fuzzy decision rules from spatial data that simultaneously have two types of uncertainties, roughness and fuzziness. Fuzzy entropy and fuzzy cross entropy are used to measure accuracies of the fuzzy decisions on unseen objects using the rules extracted. An example of neural tube birth defects is given in this paper. The identification result from rough fuzzy sets based model was compared with those from two classical rule extraction methods and three commonly used fuzzy set based rule extraction models. The comparison results support that the rule extraction model established is effective in dealing with spatial data which have roughness and fuzziness simultaneously. [Copyright &y& Elsevier]

Real life data sets often suffer from missing data. The neuro-rough-fuzzy systems proposed hitherto often cannot handle such situations. The paper presents a neuro-fuzzy system for data sets with missing values. The proposed solution is a complete neuro-fuzzy system. The system creates a rough fuzzy model from presented data (both full and with missing values) and is able to elaborate the answer for full and missing data examples. The paper also describes the dedicated clustering algorithm. The paper is accompanied by results of numerical experiments. [ABSTRACT FROM AUTHOR]

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft–rough fuzzy sets, which extends Dubois and Prade’s rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory. [ABSTRACT FROM AUTHOR]