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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 $w$.

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

Let $<$ be a fixed monomial ordering on $T$. We say that 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 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 $d$

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.


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