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 context
with 
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
.