Abstract:
This paper describes and analyzes the stilted buildings of the Tujia people (an ethnic group living in mainland China), a distinctive building style unique to them, from the perspectives of site selection, spatial layout, construction techniques, and cultural inheritance. The cluster of stilted buildings (Diaojiao Lou in Mandarin Pinyin) in the Pengjia Village (meaning most of the villagers share the surname of Peng) is presented as a case study in this paper. The paper makes a case for their preservation as authentic carriers of the Tujia people’s cultural history, which is quickly disappearing due to development pressures. Three preservation strategies are discussed to meet this preservation goal. The first is to provide a detail analysis of the construction language to guarantee authenticity in the documentation, preservation and restoration processes of the stilted buildings. The second is to keep alive the expert knowledge and skill of traditional artisans by involving them in the construction of new structures using diaojiaolou techniques. The third strategy is to encourage local people to “dress-up” discordant buildings constructed mid to late 20th century with well-mannered facades using traditional details such as suspension columns, shuaqi, and six-panel and bang doors. Taking as a whole, these strategies are presented to help local residents, preservation experts, developers and policy makers sustain the irreplaceable cultural heritage and economic independence of the Tujia people.

Abstract:
Covering-based rough set theory is a useful tool to deal with inexact, uncertain, or vague knowledge in information systems. Geometric lattice has been widely used in diverse fields, especially search algorithm design, which plays an important role in covering reductions. In this paper, we construct three geometric lattice structures of covering-based rough sets through matroids and study the relationship among them. First, a geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by a covering. Then its characteristics, such as atoms, modular elements, and modular pairs, are studied. We also construct a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, we present three sufficient and necessary conditions for two types of covering upper approximation operators to be closure operators of matroids. We also represent two types of matroids through closure axioms and then obtain two geometric lattice structures of covering-based rough sets. Third, we study the relationship among these three geometric lattice structures. Some core concepts such as reducible elements in covering-based rough sets are investigated with geometric lattices. In a word, this work points out an interesting view, namely, geometric lattice, to study covering-based rough sets. 1. Introduction Rough set theory [1] was proposed by Pawlak to deal with granularity in information systems. It is based on equivalence relations. However, the equivalence relation is rather strict, hence the applications of the classical rough set theory are quite limited. For this reason, rough set theory has been extended to generalized rough set theory based on tolerance relation [2], similarity relation [3], and arbitrary binary relation [4–8]. Through extending a partition to a covering, we generalize rough set theory to covering-based rough set theory [9–11]. Because of its high efficiency in many complicated problems such as attribute reduction and rule learning in incomplete information/decision, covering-based rough set theory has been attracting increasing research interest [12, 13]. Lattice is suggested by the form of the Hasse diagram depicting it. In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). They encode the algebraic behavior of the entailment relation and such basic logical connectives as

Abstract:
The theory of rough sets is concerned with the lower and upper approximations of objects through a binary relation on a universe. It has been applied to machine learning, knowledge discovery, and data mining. The theory of matroids is a generalization of linear independence in vector spaces. It has been used in combinatorial optimization and algorithm design. In order to take advantages of both rough sets and matroids, in this paper we propose a matroidal structure of rough sets based on a serial and transitive relation on a universe. We define the family of all minimal neighborhoods of a relation on a universe and prove it satisfies the circuit axioms of matroids when the relation is serial and transitive. In order to further study this matroidal structure, we investigate the inverse of this construction: inducing a relation by a matroid. The relationships between the upper approximation operators of rough sets based on relations and the closure operators of matroids in the above two constructions are studied. Moreover, we investigate the connections between the above two constructions. 1. Introduction The theory of rough sets [1] proposed by Pawlak is an extension of set theory for handling incomplete and inexact knowledge in information and decision systems. And it has been successfully applied to many fields, such as machine learning, granular computing [2], data mining, approximate reasoning, attribute reduction [3–6], and rule induction [7, 8]. For an equivalence relation on a universe, a rough set is a formal approximation of a crisp set in terms of a pair of sets which give the lower and the upper approximations of the original set. In order to meet many real applications, the rough sets have been extended to generalized rough sets based on relations [9–14] and covering-based rough sets [15–17]. In this paper, we focus on generalized rough sets based on relations. Matroid theory [18] proposed by Whitney is a generalization of linear algebra and graph theory. And matroids have been used in diverse fields, such as combinatorial optimization, algorithm design, information coding, and cryptology. Since a matroid can be defined by many different but equivalent ways, matroid theory has powerful axiomatic systems. Matroids have been connected with other theories, such as rough sets [19–22], generalized rough sets based on relations [23, 24], covering-based rough sets [25, 26] and lattices [27–29]. Rough sets and matroids have their own application fields in the real world. In order to make use of both rough sets and matroids, researchers have combined

Abstract:
Covering is a type of widespread data representation while covering-based rough sets provide an efficient and systematic theory to deal with this type of data. Matroids are based on linear algebra and graph theory and have a variety of applications in many fields. In this paper, we construct two types of covering cycle matroids by a covering and then study the graphical representations of these two types of matriods. First, through defining a cycle graph by a set, the type-1 covering cycle matroid is constructed by a covering. By a dual graph of the cycle graph, the covering can also induce the type-2 covering cycle matroid. Second, some characteristics of these two types of matroids are formulated by a covering, such as independent sets, bases, circuits, and support sets. Third, a coarse covering of a covering is defined to study the graphical representation of the type-1 covering cycle matroid. We prove that the type-1 covering cycle matroid is graphic while the type-2 covering cycle matroid is not always a graphic matroid. Finally, relationships between these two types of matroids and the function matroid are studied. In a word, borrowing from matroids, this work presents an interesting view, graph, to investigate covering-based rough sets. 1. Introduction Covering is a type of common and important data organization mode, and it most appears in incomplete information/decision systems based on symbolic data [1, 2], numeric and fuzzy data [3, 4]. Covering-based rough set theory [5, 6] is an efficient tool to process these types of data. Recently, this theory has attracted much research interest with fruitful achievements on both theory and applications. For example, it has been applied to build axiomatic systems [7, 8] and establish knowledge reduction approaches [9, 10]. Moreover, it also has been used to construct covering structures [11, 12] and define minimal covering reducts [13, 14]. However, this theory has its own limitation in dealing with some hard problems including knowledge reduction. In order to improve its ability to process those hard problems, some other mathematical theories, such as fuzzy set theory [15, 16], topology [5, 17], Boolean algebra [18, 19], and matroid [20, 21] have been combined with covering-based rough set theory. Matroid theory [22] proposed by Whitney is a generalization of linear algebra, graph theory, and transcendence theory. The original purpose of this theory is to formalize the similarities between the ideas of independence and rank in graph theory and those of linear independence and dimension in the study of

Abstract:
Rough set theory is a useful tool to deal with uncertain, granular and incomplete knowledge in information systems. And it is based on equivalence relations or partitions. Matroid theory is a structure that generalizes linear independence in vector spaces, and has a variety of applications in many fields. In this paper, we propose a new type of matroids, namely, partition-circuit matroids, which are induced by partitions. Firstly, a partition satisfies circuit axioms in matroid theory, then it can induce a matroid which is called a partition-circuit matroid. A partition and an equivalence relation on the same universe are one-to-one corresponding, then some characteristics of partition-circuit matroids are studied through rough sets. Secondly, similar to the upper approximation number which is proposed by Wang and Zhu, we define the lower approximation number. Some characteristics of partition-circuit matroids and the dual matroids of them are investigated through the lower approximation number and the upper approximation number.

Abstract:
Recently, in order to broad the application and theoretical areas of rough sets and matroids, some authors have combined them from many different viewpoints, such as circuits, rank function, spanning sets and so on. In this paper, we connect the second type of covering-based rough sets and matroids from the view of closure operators. On one hand, we establish a closure system through the fixed point family of the second type of covering lower approximation operator, and then construct a closure operator. For a covering of a universe, the closure operator is a closure one of a matroid if and only if the reduct of the covering is a partition of the universe. On the other hand, we investigate the sufficient and necessary condition that the second type of covering upper approximation operation is a closure one of a matroid.

Abstract:
In covering based rough sets, the neighborhood of an element is the intersection of all the covering blocks containing the element. All the neighborhoods form a new covering called a covering of neighborhoods. In the course of studying under what condition a covering of neighborhoods is a partition, the concept of repeat degree is proposed, with the help of which the issue is addressed. This paper studies further the application of repeat degree on coverings of neighborhoods. First, we investigate under what condition a covering of neighborhoods is the reduct of the covering inducing it. As a preparation for addressing this issue, we give a necessary and sufficient condition for a subset of a set family to be the reduct of the set family. Then we study under what condition two coverings induce a same relation and a same covering of neighborhoods. Finally, we give the method of calculating the covering according to repeat degree.

Abstract:
Granular association rule is a new approach to reveal patterns hide in many-to-many relationships of relational databases. Different types of data such as nominal, numeric and multi-valued ones should be dealt with in the process of rule mining. In this paper, we study multi-valued data and develop techniques to filter out strong however uninteresting rules. An example of such rule might be "male students rate movies released in 1990s that are NOT thriller." This kind of rules, called negative granular association rules, often overwhelms positive ones which are more useful. To address this issue, we filter out negative granules such as "NOT thriller" in the process of granule generation. In this way, only positive granular association rules are generated and strong ones are mined. Experimental results on the movielens data set indicate that most rules are negative, and our technique is effective to filter them out.

Abstract:
The expansion axiom of matroids requires only the existence of some kind of independent sets, not the uniqueness of them. This causes that the base families of some matroids can be reduced while the unions of the base families of these matroids remain unchanged. In this paper, we define unique expansion matroids in which the expansion axiom has some extent uniqueness; we define union minimal matroids in which the base families have some extent minimality. Some properties of them and the relationship between them are studied. First, we propose the concepts of secondary base and forming base family. Secondly, we propose the concept of unique expansion matroid, and prove that a matroid is a unique expansion matroid if and only if its forming base family is a partition. Thirdly, we propose the concept of union minimal matroid, and prove that unique expansion matroids are union minimal matroids. Finally, we extend the concept of unique expansion matroid to unique exchange matroid and prove that both unique expansion matroids and their dual matroids are unique exchange matroids.

Abstract:
Recommender systems are popular in e-commerce as they suggest items of interest to users. Researchers have addressed the cold-start problem where either the user or the item is new. However, the situation with both new user and new item has seldom been considered. In this paper, we propose a cold-start recommendation approach to this situation based on granular association rules. Specifically, we provide a means for describing users and items through information granules, a means for generating association rules between users and items, and a means for recommending items to users using these rules. Experiments are undertaken on a publicly available dataset MovieLens. Results indicate that rule sets perform similarly on the training and the testing sets, and the appropriate setting of granule is essential to the application of granular association rules.