Difference between revisions of "ApCoCoA-1:NC.SetOrdering"
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<title>NC.SetOrdering</title> | <title>NC.SetOrdering</title> | ||
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| − | Set | + | Set a word ordering on the monoid of all words in a non-commutative polynomial ring. |
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<syntax> | <syntax> | ||
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<types> | <types> | ||
<type>non_commutative</type> | <type>non_commutative</type> | ||
</types> | </types> | ||
| − | <key> | + | <key>ncpoly.SetOrdering</key> |
<key>NC.SetOrdering</key> | <key>NC.SetOrdering</key> | ||
<key>SetOrdering</key> | <key>SetOrdering</key> | ||
| − | <wiki-category> | + | <wiki-category>Package_ncpoly</wiki-category> |
</command> | </command> | ||
Revision as of 14:49, 25 April 2013
NC.SetOrdering
Set a word ordering on the monoid of all words in a non-commutative polynomial ring.
Syntax
NC.SetOrdering(Ordering:STRING)
Description
Note that the default ordering is "LLEX" (length-lexicographic ordering).
@param Ordering: a string which indicates an (admissible) ordering. For the time being, the package supports "LLEX" (length-lexicographic ordering), "ELIM" (elimination ordering) and "LRLEX" (length-reverse-lexicographic ordering).
Let X=x_{1}x_{2}...x_{n}. We define the (left-to-right) lexicographic ordering "LEX" on <X> as follows. For two words W1, W2 in <X>, we say W1>_{Lex}W2 if we have W1=W2*W for some non-empty word W in <X>, or if we have W1=W*x_{i}*W3, W2=W*x_{j}*W4 for some words W,W3,W4 in <X> and some letters x_{i},x_{j} in X such that i<j. Thus, we have x_{1}>_{LEX}x_{2}>_{LEX}...>_{LEX}x_{n}. Note that "LEX" is not an admissible ordering on <X>. We define admissible orderings "LLEX", "ELIM" and "LRLEX" on <X> as follows.
"LLEX": for two words W1, W2 in <X>, we say W1>_{LLEX}W2 if len(W1)>len(W2), or len(W1)=len(W2) and W1 is lexicographically larger than W2.
"ELIM": it first compares the associated commutative terms lexicographically and then breaks ties using the non-commutative lexicographic orderin with respect to x_{1}>...>x_{n}. That is, for two words W1, W2 in <X>, we say W1>_{ELIM}W2 if W1 is lexicographically larger than W2 by considering W1, W2 as two terms in the commutative case, or W1=W2 by considering W1, W2 as two terms in the commutative case and W1>_{Lex}W2 (W1 is left-to-right lexicographically larger than W2 by considering W1, W2 as two words in the non-commutative case). Thus, the elimination ordering "ELIM" first eliminates the letter x_{1}, and then x_{2}, and then x_{3}, and so on and so forth.
"LRLEX": for two words W1, W2 in <X>, we say W1>_{LRLEX}W2 if len(W1)>len(W2), or len(W1)=len(W2) and W1 is larger than W2 by right-to-left lexicographic ordering.
An admissible ordering on is called length compatible if len(W1)>len(W2) implies W1 is larger than W2 for all W1, W2 in <X>. For instance, "LLEX" and "LRLEX" are length compatible while "ELIM" is not.
Example
NC.RingEnv(); Coefficient ring : Q Ordering : LLEX ------------------------------- NC.SetOrdering(<quotes>ELIM</quotes>); NC.RingEnv(); Coefficient ring : Q Ordering : ELIM -------------------------------
