... Reinert1
The author was supported by the Deutsche Forschungsgemeinschaft (DFG).
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... precedence2
By a precedence on an alphabet we mean a partial ordering on its letters.
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... ideal3
Notice that the linear combinations in these definitions of ideals in fact describe elements of . Since the elements and can be interpreted as elements of the multiplication is well-defined and gives us an element say hi in , as does again the multiplcation .
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... presentations4
A survey on groups allowing convergent presentations can be found in [MaOt89].
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... ordering5
By the usual ordering on we mean .
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...
I.e. tex2html_wrap_inline$= (+ - 1)/2$.
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... terminate6
This can also be seen, since for m we have .
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...7
E.g. we get .
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... in8
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... set9
see [Si94] page 503 for the concrete computation.
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... basis10
This set is the union of the qc-saturating sets , , , and it is a Gröbner basis as all s-polynomials reduce to zero.
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... basis11
While interreduction in group rings can destroy the property of being a Gröbner basis for certain reductions, qc-reduction allows to incorporate this idea into Gröbner basis computations and produces unique monic reduced Gröbner bases.
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... bounded12
In case ak is bounded we can still use negative powers of ak in the computations, as from the point of view of the collection process it does not matter, at what time the power rules for ak are applied. We only have to take into account that in the definition of w1 = ak-lk + ik the normal form looks different in case -lk + ik < 0.
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... ring13
compare page .
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... bounded14
In case ak is bounded we can still use negative powers of ak in the computations, as from the point of view of the collection process it does not matter, at what time the power rules for ak are applied.
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... convergent15
For a proof of this see section 8.
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... none16
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...
none'' in this context means that no reduction based on commutative divisors and using right multiples exists.
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