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7.9.3 Groebner bases for two-sided ideals in free associative algebras

We say that a monomial $v$ divides (two-sided or bilaterally) a monomial $w$, if there exist monomials $p,s \in X$, such that $w = p \cdot v \cdot s$, in other words $v$ is a subword of $w$.

Let $T := K\langle x_1,\dots,x_n \rangle$ be the free algebra and $<$ be a fixed monomial ordering on $T$.

For a subset $G \subset K\langle x_1,\dots,x_n \rangle$, define the leading ideal of $G$ to be the two-sided ideal $LM(G) \; = \; {}_{T} \langle$ $\; \{lm(g) \;\vert\; g \in G\setminus\{0\} \}$ $\; \rangle_{T} \subseteq T$.

A subset $G\subset I$ is a (two-sided) Groebner basis for the ideal $I$ with respect to $<$, if $LM(G) = LM(I)$.

That is $\forall f\in I\setminus\{0\}$ there exists $g\in G$, such that $lm(g)$ divides $lm(f)$.

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 $d$

results in the truncated Groebner basis $G(d)$. In other words, by trimming elements of degree exceeding $d$ from the complete Groebner basis $G$, one obtains precisely $G(d)$.

In general, given a set $G(d)$, which is the result of Groebner basis computation up to weighted degree bound $d$, then it is the complete finite Groebner basis, if and only if $G(2d-1)=G(d)$ 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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