Difference between revisions of "ApCoCoA-1:NCo.SetOrdering"
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</syntax> | </syntax> | ||
<description> | <description> | ||
| − | Note that each word ordering is induced by the order of letters in X (see <ref>NCo.SetX</ref>). For instance, | + | Note that each word ordering is induced by the order of letters in X (see <ref>NCo.SetX</ref>). For instance, |
| + | NCo.SetX("abcdef"); | ||
| + | NCo.SetOrdering("ELIM"); | ||
| + | defines an elimination ordering induced by a>b>b>d>e>f. | ||
<example> | <example> | ||
NCo.RingEnv(); | NCo.RingEnv(); | ||
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</description> | </description> | ||
<seealso> | <seealso> | ||
| − | |||
<see>NCo.SetX</see> | <see>NCo.SetX</see> | ||
</seealso> | </seealso> | ||
<types> | <types> | ||
| − | |||
<type>non_commutative</type> | <type>non_commutative</type> | ||
</types> | </types> | ||
Revision as of 15:45, 30 April 2013
NCo.SetOrdering
Set a word ordering on <X>.
Note that a word ordering is a well-ordering which is compatible with multiplication. The default ordering is "LLEX" (the length-lexicographic ordering).
Let X={x_{1}x_{2}...x_{n}}. We define the non-commutative (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 a word ordering on <X>. We define word 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 ordering with respect to x_{1}>_{LEX}...>_{LEX}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 them as two terms in the commutative case, or W1=W2 by considering them as two terms in the commutative case and W1>_{Lex}W2 where "LEX" is the non-commutative left-to-right lexicographic ordering. 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": we say W>_{LRLEX}W' if len(W)>len(W'), or len(W)=len(W') and W is larger than W' by the non-commutative right-to-left lexicographic ordering.
A word ordering on is said to be 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.
Syntax
NCo.SetOrdering(Ordering:STRING)
Description
Note that each word ordering is induced by the order of letters in X (see NCo.SetX). For instance,
NCo.SetX("abcdef");
NCo.SetOrdering("ELIM");
defines an elimination ordering induced by a>b>b>d>e>f.
Example
NCo.RingEnv(); Coefficient ring : Q Ordering : LLEX ------------------------------- NCo.SetOrdering(<quotes>ELIM</quotes>); NCo.RingEnv(); Coefficient ring : Q Ordering : ELIM -------------------------------
See also
