
Operadic Homological Algebra and Gröbner Bases 197
where the second term is immediately reduced to
(−1)
N(j−1)+N(i−1)+N (i−1)
µ
N
◦
N
(µ
2
◦
2
(µ
N
◦
N
(µ
2
◦
2
µ
N
))) =
= (−1)
N(j−1)
µ
N
◦
N
(µ
2
◦
2
(µ
N
◦
N
(µ
2
◦
2
µ
N
))),
using the relation R
i,1
twice, while the first term is reduced to the same result
through a lengthier sequence of reductions (depending on i+j being less than,
equal to, or greater than N).
The small common multiple (µ
N
◦
i
(µ
2
◦
2
µ
N
))◦
i
µ
2
of the leading monomial
of the relation R
i,1
with the leading monomial of the left-hand side of (6.1)
creates an S-polynomial that can be reduced to R
i,2
, and hence can be reduced
to zero using our candidate for the reduced Gröbner basis. ...