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
.
Let
be the free algebra and
$<$ be a fixed monomial ordering on $T$.
For a subset
,
define the leading ideal of to be the two-sided ideal
.
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.
Note: If the set of input polynomials is not weighted homogeneous with respect to the
weights of the ring variables, and a Groebner is not finite,
then actually not much can be said precisely on the properties of the given ideal.
By increasing the length bound bigger generating sets will be computed, but in contrast to the
weighted homogeneous case some polynomials in of small length first enter the basis after
computing up to a much higher length bound.
|