... bases¹

Note that similar concepts appear in a paper of Hironaka where the notion of a complete set of polynomials is called a standard basis [Hi64].

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... admissible²

A term ordering $\succeq$ is called admissible if for every term s,t,u, $s \succeq 1$ holds, and $s \succeq t$ implies $s \circ u \succeq t \circ u$. An ordering fulfilling the latter condition is also said to be compatible with the respective multiplication $\circ$.

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... ring³

${\bf Q}$ denotes the rational numbers.

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... zero⁴

Note that we always assume that the reduction in the ring is effective.

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... multiplication⁵

When studying monoid rings over reduction rings it is possible that the ordering on the ring is not compatible with scalar multiplication as well as with multiplication with monomials or polynomials.

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... effective⁶

By ``effective'' we mean that given an element we can decide whether a successor exists and then construct it.

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...$T \subseteq \Sigma^* \times \Sigma^*$⁷

$\Sigma^*$ is the set of all words on the alphabet $\Sigma$ where $\lambda$ presents the empty word, i.e., the word of length zero, and $\equiv$ the identity on words.

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... admissible⁸

An ordering on $\Sigma^*$ is called admissible, if for all $u, v, w, x, y \in \Sigma^*$, $w \succeq \lambda$ holds and $u \succ v$ implies $xuy \succ xvy$.

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... polynomials⁹

Note that we use $1 = 1 \cdot\lambda = \lambda$.

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... divisibility¹⁰

We call a term t (right) divisible by a term x in case there exists a term z such that $t = x \circ z$.

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... polynomials¹¹

Notice that p₁=p₂ is possible.

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... shows¹²

This property is important for introducing interreduction to a completion procedure.

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... non-constant¹³

A constant polynomial is an element in ${\bf K}$.

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... none¹⁴

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...

By theorem 2.2 the existence of such finite bases would solve the subgroup problem for groups presented by convergent semi-Thue systems.

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