
42 Algebraic Operads: An Algorithmic Companion
The maximal weight of a leading monomial is 3, so V = {x
2
, xz, zy, yz}, and
the graph of normal forms is
x
2
xz
//
zy yz
We conclude that the set of reduced monomials is finite, and consists of 9
elements 1, x, y, z, x
2
, xz, zy, yx, xzy. An interesting remark made in [252] is
that if we forget about the normal word 1, and view the equations xy = z,
yz = x, zx = y as defining relations of a semigroup (rather than monoid),
we can easily conclude that the given relations give a presentation of Q
8
as a semigroup by generators and relations. (There is a natural surjective
map from this semigroup to the quaternion ...