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
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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 978-1-60566-324-1.ch016.m01 be a set of objects, 978-1-60566-324-1.ch016.m02is a set of attributes, and 978-1-60566-324-1.ch016.m03 is a relation between 978-1-60566-324-1.ch016.m04 and 978-1-60566-324-1.ch016.m05. 978-1-60566-324-1.ch016.m06 is called a formal context (context for short). Let 978-1-60566-324-1.ch016.m07 is a subset of 978-1-60566-324-1.ch016.m08and 978-1-60566-324-1.ch016.m09 is a subset of978-1-60566-324-1.ch016.m10,we define two functions978-1-60566-324-1.ch016.m11 and 978-1-60566-324-1.ch016.m12 as:978-1-60566-324-1.ch016.m13978-1-60566-324-1.ch016.m14then, 978-1-60566-324-1.ch016.m15 is called a formal concept (concept for short) on context978-1-60566-324-1.ch016.m16with 978-1-60566-324-1.ch016.m17 and 978-1-60566-324-1.ch016.m18, where 978-1-60566-324-1.ch016.m19, 978-1-60566-324-1.ch016.m20. 978-1-60566-324-1.ch016.m21 is called extent of the concept, 978-1-60566-324-1.ch016.m22 is called intent of the concept. The set of all concepts on 978-1-60566-324-1.ch016.m23 is denoted by 978-1-60566-324-1.ch016.m24

If978-1-60566-324-1.ch016.m25,978-1-60566-324-1.ch016.m26, there are some properties, which will be used in this chapter, as follows:

  • 978-1-60566-324-1.ch016.m27978-1-60566-324-1.ch016.m28

  • 978-1-60566-324-1.ch016.m29978-1-60566-324-1.ch016.m30

  • 978-1-60566-324-1.ch016.m31978-1-60566-324-1.ch016.m32

  • 978-1-60566-324-1.ch016.m33978-1-60566-324-1.ch016.m34

Note that 4) and 4') can be extended as follows: given by an index set 978-1-60566-324-1.ch016.m35, if for each 978-1-60566-324-1.ch016.m36, 978-1-60566-324-1.ch016.m37and978-1-60566-324-1.ch016.m38, then

978-1-60566-324-1.ch016.m39.

On the other hand, by 2) and 2’), for any subset978-1-60566-324-1.ch016.m40 of 978-1-60566-324-1.ch016.m41, 978-1-60566-324-1.ch016.m42 must be a concept and for any subset 978-1-60566-324-1.ch016.m43of 978-1-60566-324-1.ch016.m44, 978-1-60566-324-1.ch016.m45 must be a concept as well.Especially, for an object 978-1-60566-324-1.ch016.m46,978-1-60566-324-1.ch016.m47is called object concept and is denoted by 978-1-60566-324-1.ch016.m48, for a attribute 978-1-60566-324-1.ch016.m49, 978-1-60566-324-1.ch016.m50is called attribute concept and is denoted by978-1-60566-324-1.ch016.m51.

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