A coherent configuration on a finite set 
is a set partition 
, 
, ..., 
of 
. Each 
is a binary relation, meaning
a subset of 
.
The diagonal relation 
is a union of the 
, and the inverse (or transpose)
of every relation is another relation. Furthermore, whenever 
, the number
 |
depends only on 
,
,
and 
.
The constants 
are the intersection numbers
of the configuration.
A coherent configuration is homogeneous when the diagonal itself is one relation; homogeneous coherent configurations are association
schemes. Its adjacency matrices span an adjacency
algebra, which in the homogeneous case is the Bose-Mesner
algebra. A coherent configuration is called Schurian
when its relations are the orbitals of a permutation
group acting on 
.
