A Minimal Family of Sub-bases

Abstract

This paper investigates a minimal family of sub-bases. First, the concept of a minimal family of sub-bases is presented and its properties are studied. Then the relationship between reducts in covering information systems and minimal families of sub-bases is discussed. Based on Boolean matrices, an approach is provided to derive a minimal family of sub-bases. Finally, experiments are conducted to illustrate the effectiveness of the proposed approach.Mathematics Subject Classification (2010). 54A05, 54B15, 54C05, 54C10 homomorphisms or mappings between two information systems gains more attention in recent years. The motivation of study homomorphisms or mappings between information systems is to find a relatively small information system which has the same reduct as the original database [16]. initially introduced the concept of homomorphism, which is used as a tool to study the relationship between information systems based on rough set. Li and Ma [5] studied some features of redundancy and reduct of complete information systems under some homomorphisms. Later, many authors [2,3,5,9,[12][13][14][15][16][17][18][19][20][23][24][25][26][27][28] discussed homomorphisms or mappings between information systems based on rough set. A consistent function related to coverings was proposed by Wang et al. [16]. By analyzing the consistent function related to coverings, we find that the work to structure a consistent function is a process to seek a representation element under some conditions. This process is similar to structuring a quotient space under an equivalence relation in a topological space. Hence, this paper explains consistent functions related to coverings and homomorphisms from the perspective of topology.The remainder of this paper is organized as follows. In Section 2, the definition of a minimal family of sub-bases is presented, and its properties are investigated. Section 3 discusses the relationship between reducts and minimal families of sub-bases. Based on Boolean matrices, Section 4 proposes an approach to derive a minimal family of sub-bases. In Section 5, several numerical experiments are conducted on UCI data sets to evaluate the proposed method. Section 6 has some concluding remarks. A minimal family of sub-basesSuppose S i is a sub-base for finite topological space (X, τ i ) for i = 1, 2, . . . , n, ∆ = {S 1 , S 2 , . . . , S n }, and S ∆ = 1, 2, . . . , n}. Then S ∆ is a sub-base for a topology τ ∆ of finite set X. For each subfamily ∆ ′ of ∆, a question is: are the topologies generated by both S ∆ and S ∆ ′ as sub-bases the same? Now we present the definition of a minimal family of sub-bases, which keeps the topology unchanged.Definition 2.1. Let S i be a sub-base for finite topological space (X, τ i ) for i = 1, 2, . . . , n and ∆ = {S 1 , S 2 , . . . , S n }. If the following statement holds, then the family ∆ of subbases is called a minimal family of sub-bases with respect to X. The statement is: for any subfamily ∆ ′ of ∆, if S ∆ ′ is a sub-base for finite topological space (X, τ ∆ ), then ∆...