7.9.3 Groebner bases for two-sided ideals in free associative algebras
We say that a monomial
divides (two-sided 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 two-sided ideal
.
Let be a fixed monomial ordering on .
We say that a subset is a (two-sided) Groebner basis for the ideal with respect to , if . That is
there exists , such that
divides .
The notion of Groebner-Shirshov basis applies to more general algebraic structures,
but means the same as Groebner basis for associative algebras.
Suppose, that the weights of the ring variables are strictly positive.
We can interprete these weights as defining a non-standard 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.
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