In simple terms, let 
, 
, and 
be members of an algebra. Then the algebra is said to be associative if
 | (1) |
where 
denotes multiplication. More formally, let 
denote an 
-algebra, so that 
is a vector space over 
and
 | (2) |
 | (3) |
Then 
is said to be 
-associative if there exists an 
-dimensional subspace 
of 
such that
 | (4) |
for all 
and 
. Here, vector multiplication 
is assumed to be bilinear. An 
-dimensional 
-associative algebra is simply said to be "associative."
