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
.