Invariant characters of information systems under some homomorphisms
Introduction
The concept of an information system (also called a knowledge representation system), with manipulations, based on rough set theory was introduced by Pawlak in [1]. This concept and its approach arose from studying problems of expert system, inductive reasoning, learning theory, cluster analysis, and so on. Many topics on an information system have been widely investigated by many scholars (see [2], [3], [4], [5], [6]).
The notion of a homomorphism of information systems as a kind of tool to study the relationship between two information systems was introduced by Grzymala-Busse in [3]. In [5], the authors depicted the conditions which make an information system S=(U,Q,V,f) to be selective (i.e., for arbitrary ) in terms of endomorphisms of S. The concept of a homomorphism on information systems is very useful for aggregating sets of objects, attributes, and descriptors of the original system.
In this paper, our main aim is to discuss some invariant characters of an information system under a homomorphism on it. At the same time, we shall give a sufficient condition guaranteeing the existence of an endomorphism of an information system.
In the following section, we recall some terms and definitions to be used in the paper. In Section 3, we establish some lemmas for the sake of the verification of our main results in Section 4. In Section 5, we employ an example to illustrate our main results. Section 6 concludes the paper.
Section snippets
Information systems
By an information system S, we mean an ordered quadruplewhere U is a set called the universe of S – elements of U are called objects, Q is a set of attributes, V(=⋃a∈QVa) is a set of values of attributes – Va will be called the domain of a, f:U×Q→V is a description function, such that f(x,a)∈Va for every a∈Q and x∈U.
We say that objects x,y∈U are indiscernible with respect to a∈Q iff f(x,a)=f(y,a), and we shall write . It is clear that ã is an equivalence relation on U. Objects x,y∈U
Lemmas
Lemma 3.1 Let S=(U,Q,V,f) and S′=(U′,Q′,V′,f′) be two information systems and (ho,hA,hD) a homomorphism of S into S′. Then ho([x]P)⊆[ho(x)]hA(P) for arbitrary subset P⊆Q and arbitrary x∈U. In practice, if ho is a surjection and hD is one-to-one, then the inverse inclusion is available, namely, ho([x]P)=[ho(x)]hA(P). Proof For arbitrary y′∈ho([x]P), there exists an element y∈[x]P such that ho(y)=y′. By the definition of a homomorphism, for any a∈P, it follows that f′(ho(y),hA(a))=hD(f(y,a))=hD(f(x,a))=f′(ho(x),hA
Theorems
Theorem 4.1 Let S=(U,Q,V,f) be an information system. Let P⊆Q, Q−P be superfluous in Q, and for each a∈Q−P, there exists a′∈P such that , namely, a′ dependent on a. Denote the information system (U,P,∪a∈PVa,f′) by S′, where f′ is the restriction of f to set U×P. Then, there exists an endomorphism h of S into S′. Proof If P=Q, it is obvious that the theorem is true. So we can assume that P⊂Q. We define a mapping h: S→S′ as follows:where
An example
In this section, we employ an example to illustrate our main results.
Let S=(U,Q,V,f) and S′=(U′,Q′,V′,f′) be two information systems described in Table 3, Table 4, respectively.
Let ho be a mapping from U onto U′ defined as following:
Let hA be a mapping from Q to Q′ defined as following:
Let hD be a mapping from V to V′ that consists of . They are defined by following:
It is very easy to verify
Conclusions
We know that the theory of rough sets deals with the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations (see [1,7]). However, lower and upper approximations are not primitive notions. They are constructed from other concepts, such as binary relations on a universe, partitions and coverings of a universe, and approximation space. For an information system, we can see it as a mixture with some approximation spaces on the
Acknowledgements
This work has been supported by the National Youth Science Foundation of China under Grant 69805004 and the Youth Science Foundation of Shanxi under Grant 981002. The authors thank the referee very much for his or her comments.
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