
286 Algebraic Operads: An Algorithmic Companion
9.3.2 Relation rank 1
The relation matrices for the five cases corresponding to relation rank 1
have been displayed in (9.4). For case 1, we substitute 1, x
1
, x
2
, x
3
, x
4
for
x
1
, x
2
, x
3
, x
4
, x
5
into (9.9) to obtain the matrix C of consequences in arity 5:
C =
1 x
1
x
2
x
3
x
4
0 0 0 0 0 0 0 0 0
1 0 x
1
0 0 x
2
0 x
3
x
4
0 0 0 0 0
0 1 0 x
1
0 0 x
2
0 0 x
3
0 x
4
0 0
0 0 1 0 x
1
0 0 x
2
0 0 x
3
0 x
4
0
0 0 0 0 0 1 x
1
0 x
2
0 0 x
3
0 x
4
0 0 0 0 0 0 0 0 0 1 x
1
x
2
x
3
x
4
(9.10)
Computing the partial Smith form using Algorithm 8.4.2.3, we obtain
C −→
I
5
O
5,9
O
1,5
B
1,9
where the row vector B = B
1,9
contains the following nonzero entries, which
ha