32 Algebraic Operads: An Algorithmic Companion
This is a diamond-shaped diagram that gave the name to the corresponding
formalism. Informally, in order to extend S to a Gröbner basis, one must
look for ambiguities and, in case two different reductions lead to two different
reduced expressions of the same element, we adjoin the difference of those
reduced expressions to S (“resolve the ambiguity”), making the new set a
more plausible candidate for a Gröbner basis. In other words, “to compute
a Gröbner basis, one must ensure that all rewriting diagrams close up into
diamonds”.
Definition 2.4.1.1 (S-polynomial). Let g
1
, g
2
∈ T (X) be two monic polyno-
mials. Assume that for some monomials u
1
, u
2
, v which are all different from 1,
we have lm(g
1
) = u
1
v and lm