... Reinert¹

The author was supported by the Deutsche Forschungsgemeinschaft (DFG).

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... admissible²

A partial ordering $\succeq$ on $\Sigma^*$ is called admissible if for all u,v,x,y in $\Sigma^*$ we have $u \succeq \lambda$, and $u \succ v$ implies $xuy \succ xvy$.

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... well-founded³

A partial ordering $\succeq$ on $\Sigma^*$ is called well-founded if no infinite chains of the form $x_1 \succ x_2 \succ \ldots$ with $x_i \in \Sigma^*$ are possible.

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... fair⁴

A fair strategy will ensure that all elements of the set B are considered at some time by the procedure.

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...

These polynomials are frequently called s-polynomials in the literature.

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...${\sf ideal}_{r}^{}(F) = \{ \sum_{i=1}^n \alpha_i \cdot f_i \ast w_i \mid n \in {\bf N}, \alpha_i \in {\bf K}, f_i \in F, w_i \in {\cal M}\}$⁵

${\bf N}$ denotes the natural numbers including 0.

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... closure⁶

The normal closure of a set T in ${\cal F}_{\Sigma}$ is the smallest normal subgroup containing T.

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... none⁷

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...

By theorem 6 the existence of such finite bases would solve the word problem for groups presented by finite string rewriting systems.

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