... 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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A term ordering is called admissible if for every term s,t,u, holds, and implies . An ordering fulfilling the latter condition is also said to be compatible with the respective multiplication .
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... ring3
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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...7
is the set of all words on the alphabet where presents the empty word, i.e., the word of length zero, and the identity on words.
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An ordering on is called admissible, if for all , holds and implies .
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... polynomials9
Note that we use .
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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 .
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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 .
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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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