7.9.3 Groebner bases for twosided ideals in free associative algebras
We say that a monomial
divides (twosided or bilaterally) a monomial
, if there exist monomials
, such that
, in other words
is a subword of
.
For a subset
,
define the leading ideal of to be the twosided ideal
.
Let be a fixed monomial ordering on .
We say that a subset is a (twosided) Groebner basis for the ideal with respect to , if . That is
there exists , such that
divides .
Suppose, that the weights of the ring variables are strictly positive.
We can interprete these weights as defining a nonstandard grading on the ring.
If the set of input polynomials is weighted homogeneous with respect to the given
weights of the ring variables, then computing up to a weighted degree (and thus, also length) bound
results in the truncated Groebner basis
. In other words, by trimming elements
of degree exceeding
from the complete Groebner basis
, one obtains precisely
.
In general, given a set
, which is the result of Groebner basis computation
up to weighted degree bound
, then
it is the complete finite Groebner basis, if and only if
holds.
