Granular Computing in Formal Concept

Granular Computing in Formal Concept

Yuan Ma (University of Science and Technology Liaoning, China), Zhangang Liu (University of Science and Technology Liaoning, China) and Xuedong Zhang (University of Science and Technology Liaoning, China)
DOI: 10.4018/978-1-60566-324-1.ch016

Abstract

Granular computing has permeated through the field of formal concept; it is another new and rapid developmental aspect of formal concept. In this chapter, we’ll regard supremum semisublattice, infimum semisublattice and sublattice as “granule”. When a set of granules covers the lattice, “granular space” is called on the concept lattice. We study mainly granular spaces generated by ideal-filter congruence relations and tolerance relations. We emphasize properties of these granular spaces and generating methods of these granular spaces. By our viewpoint to study granular computing in formal concept, we find out that it shows profound relation and essence of various sublattices.
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1. Basic Definition

Definition 1.1

Let be a set of objects, is a set of attributes, and is a relation between and . is called a formal context (context for short). Let is a subset of and is a subset of,we define two functions and as:then, is called a formal concept (concept for short) on contextwith and , where , . is called extent of the concept, is called intent of the concept. The set of all concepts on is denoted by

If,, there are some properties, which will be used in this chapter, as follows:

Note that 4) and 4') can be extended as follows: given by an index set , if for each , and, then

.

On the other hand, by 2) and 2’), for any subset of , must be a concept and for any subset of , must be a concept as well.Especially, for an object ,is called object concept and is denoted by , for a attribute , is called attribute concept and is denoted by.

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