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.