
Twisted Associative Algebras and Shuffle Algebras 133
an S-polynomial of g
1
and g
2
; the common term cancels, since both g
1
and g
2
are monic.
Example 4.5.1.2. Consider the shuffle monomial 321 ∈ T
X
(1)(3). This
monomial forms two different overlaps with itself: we have
4321 = µ
{2,3,4},{1}
(321, 1) = µ
{4},{1,2,3}
(1, 321)
and
54321 = µ
{3,4,5},{1,2}
(321, 21) = µ
{4,5},{1,2,3}
(21, 321).
In this case, there is one small common multiple of weight 4, and one small
common multiple of weight 5. However, the situation might be very different,
as the shuffle monomial 312 ∈ T
X
(1)(3) demonstrates. This monomial does
not have an overlap of weight 4 with itself: if the permutation ...