
226 Algebraic Operads: An Algorithmic Companion
Definition 7.2.3.5 (Action of permutations on orders). Let σ ∈ S
k
be any
permutation of 1, . . . , k and let ≺ be any monomial order. The monomial order
≺
σ
is defined in terms of the action of S
k
permuting the coordinates of vectors
in N
k
. That is, we first set
σ · [e
1
, . . . , e
i
, . . . , e
k
] = [e
σ(1)
, . . . , e
σ(i)
, . . . , e
σ(k)
],
and then we define ≺
σ
in terms of ≺ by
v ≺
σ
w ⇐⇒ σ
−1
· v ≺ σ
−1
· w.
(To cancel the σ on ≺ we need to apply σ
−1
.) That is, given v = [e
1
, . . . , e
k
]
and w = [f
1
, . . . , f
k
] with v 6= w, let i be the least index satisfying the inequal-
ity e
σ
−1
(i)
6= f
σ
−1
(i)
. Then v ≺ w if and only if