Difference between revisions of "ApCoCoA-1:NC.SetOrdering"
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<item><quotes>LRLEX</quotes>: for two words <tt>W1, W2</tt> in <tt><X></tt>, we say <tt>W1>_{LRLEX}W2</tt> if <tt>len(W1)>len(W2)</tt>, or <tt>len(W1)=len(W2)</tt> and <tt>W1</tt> is larger than <tt>W2</tt> by right-to-left lexicographic ordering.</item> | <item><quotes>LRLEX</quotes>: for two words <tt>W1, W2</tt> in <tt><X></tt>, we say <tt>W1>_{LRLEX}W2</tt> if <tt>len(W1)>len(W2)</tt>, or <tt>len(W1)=len(W2)</tt> and <tt>W1</tt> is larger than <tt>W2</tt> by right-to-left lexicographic ordering.</item> | ||
</itemize> | </itemize> | ||
| + | An admissible ordering on is called <em>length compatible</em> if <tt>len(W1)>len(W2)</tt> implies <tt>W1</tt> is larger than <tt>W2</tt> for all <tt>W1, W2</tt> in <tt><X></tt>. For instance, <quotes>LLEX</quotes> and <quotes>LRLEX</quotes> are length compatible while <quotes>ELIM</quotes> is not. | ||
<example> | <example> | ||
NC.RingEnv(); | NC.RingEnv(); | ||
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<see>NC.Add</see> | <see>NC.Add</see> | ||
<see>NC.Deg</see> | <see>NC.Deg</see> | ||
| − | <see>NC. | + | <see>NC.FindPolynomials</see> |
<see>NC.GB</see> | <see>NC.GB</see> | ||
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<see>NC.HF</see> | <see>NC.HF</see> | ||
<see>NC.Interreduction</see> | <see>NC.Interreduction</see> | ||
<see>NC.Intersection</see> | <see>NC.Intersection</see> | ||
| + | <see>NC.IsFinite</see> | ||
<see>NC.IsGB</see> | <see>NC.IsGB</see> | ||
<see>NC.IsHomog</see> | <see>NC.IsHomog</see> | ||
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<see>NC.SetX</see> | <see>NC.SetX</see> | ||
<see>NC.Subtract</see> | <see>NC.Subtract</see> | ||
| + | <see>NC.TruncatedGB</see> | ||
<see>NC.UnsetFp</see> | <see>NC.UnsetFp</see> | ||
<see>NC.UnsetOrdering</see> | <see>NC.UnsetOrdering</see> | ||
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<see>NC.UnsetRules</see> | <see>NC.UnsetRules</see> | ||
<see>NC.UnsetX</see> | <see>NC.UnsetX</see> | ||
| + | <see>Introduction to CoCoAServer</see> | ||
</seealso> | </seealso> | ||
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<types> | <types> | ||
<type>non_commutative</type> | <type>non_commutative</type> | ||
</types> | </types> | ||
| − | |||
<key>gbmr.SetOrdering</key> | <key>gbmr.SetOrdering</key> | ||
<key>NC.SetOrdering</key> | <key>NC.SetOrdering</key> | ||
Revision as of 15:16, 8 June 2012
NC.SetOrdering
Set an admissible ordering on <X>.
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": 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 in x_{1}, and then the letter 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 -------------------------------
See also
