Abstract
Abstract Using the notion of preconcept, we generalize Pawlak’s approximation operators from a one-dimensional space to a two-dimensional space in a formal context. In a formal context, we present two groups of approximation operators in a two-dimensional space: one is aided by an equivalence relation defined on the attribute set, and another is aided by the lattice theoretical property of the family of preconcepts. In addition, we analyze the properties of those approximation operators. All these results show that we can approximate all the subsets in a formal context assisted by the family of preconcepts using the above groups of approximation operators. Some biological examples show that the two groups of approximation operators provided in this article have potential ability to assist biologists to do the phylogenetic analysis of insects.
Highlights
Formal concept analysis proposed in [1] is a mathematical thinking for conceptual data analysis and knowledge processing
For this new notion – preconcept – it has been proved [3] that the family of preconcepts can construct a lattice with their hierarchical order; Vormbrock and Wille [4] demonstrated that the idea of preconcepts enriches the theory of formal concept analysis
As a mathematical tool to deal with data analysis and knowledge discovery, the rough set theory depends on the understanding of its basic notions, that is, lower and upper approximation operators [6]
Summary
Formal concept analysis proposed in [1] is a mathematical thinking for conceptual data analysis and knowledge processing. For a formal context, the family of preconcepts provides more information than the set of formal concepts, since we know from [1,2]. Some other methods can be seen in other studies [11,12,13,14] Since both the formal concept analysis and the rough set theory are two related mathematical tools in the areas of knowledge representation and knowledge processing, some authors introduced the notion of approximation operators into formal concept analysis. This section will give two groups of approximation operators for a given formal context in the two-dimensional space O × P, so as to approximate those subsets that are not in ( ) by the elements in ( ).
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