7.9.3 Groebner bases for two-sided ideals in free associative algebras
We say that a monomial
333#333 divides (two-sided or bilaterally) a monomial
348#348, if there exist monomials
400#400, such that
401#401, in other words
333#333 is a subword of
348#348.
Let
402#402 be the free algebra and
$<$ be a fixed monomial ordering on $T$.
For a subset
403#403,
define the leading ideal of 190#190 to be the two-sided ideal
404#404
405#405
406#406.
A subset 252#252 is a (two-sided) Groebner basis for the ideal 253#253 with respect to 228#228, if 407#407.
That is
408#408 there exists 256#256, such that
409#409 divides 410#410.
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 interpret 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
171#171
results in the truncated Groebner basis
411#411. In other words, by trimming elements
of degree exceeding
171#171 from the complete Groebner basis
190#190, one obtains precisely
411#411.
In general, given a set
411#411, which is the result of Groebner basis computation
up to weighted degree bound
171#171, then
it is the complete finite Groebner basis, if and only if
412#412 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.
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