... matrices¹

These matrices can be computed during the Gröbner basis computation.

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... considered²

In fact this test set is related to the technique called saturation and explained more precisely in [5].

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...$K_p$³

Note that this ordering is well-founded since we have a well-founded ordering on $\Sigma^*$ and $K_p \in\mathbb{N}$.

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...$K_p$⁴

Note that this ordering is well-founded since we have a well-founded ordering on $\Sigma^*$ and $K_p \in\mathbb{N}$.

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... ${\sf HT}(g_{i,s}){\sf HT}(q_i) < t_p$⁵

Remember that ${\sf HT}(g_{i,s}){\sf HT}(h_i)v\leq {\sf HT}(g_{i,s} \ast w_l)v < t_p$.

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... ⁶

If not stated otherwise $<$ is the usual ordering on $\mathbb{Z}$ where $\ldots < -2 < -1 < 0 < 1 < 2 < \ldots$.

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...$a <_Z b$⁷

Let $c \in \mathbb{N}$. We call the positive numbers $0, \ldots, c-1$ the remainders of $c$. Then for each $d \in \mathbb{Z}$ there are unique $a,b \in \mathbb{Z}$ such that $d=a \cdot c + b$ and $b$ is a remainder of $c$. We get $b<c$ and in case $d>0$ and $a \not = 0$ even $c \leq d$. Further $c$ does not divide $b_1 - b_2$, if $b_1, b_2$ are different remainders of $c$.

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... $G\backslash \{ g \} \cup \{ canon(g') \}$⁸

$canon(p)=p$ if $HC(p)>0$ and $canon(p)= -p$ otherwise.

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... property⁹

By Theorem 6 a set can be characterized as a prefix Gröbner basis if any element in its right ideal can be prefix reduced to zero using the set.

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