7.9.3 Groebner bases for two-sided ideals in free associative algebras
We say that a monomial
331#331 divides (two-sided or bilaterally) a monomial
346#346, if there exist monomials
398#398, such that
399#399, in other words
331#331 is a subword of
346#346.
Let
400#400 be the free algebra and
$<$ be a fixed monomial ordering on $T$.
For a subset
401#401,
define the leading ideal of 189#189 to be the two-sided ideal
402#402
403#403
404#404.
A subset 250#250 is a (two-sided) Groebner basis for the ideal 251#251 with respect to 226#226, if 405#405.
That is
406#406 there exists 254#254, such that
407#407 divides 408#408.
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
409#409. In other words, by trimming elements
of degree exceeding
171#171 from the complete Groebner basis
189#189, one obtains precisely
409#409.
In general, given a set
409#409, 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
410#410 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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