Difference between revisions of "ApCoCoA-1:NC.SetOrdering"
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</itemize> | </itemize> | ||
<par/> | <par/> | ||
| − | In the following, we let <tt> | + | In the following, we let <tt>W^n</tt> be the monoid of all words generated by <tt>{x[1],x[2],...,x[n]}</tt>. We define the (left-to-right) lexicographic ordering <quotes>LEX</quotes> on <tt>W^n</tt> as follows. For two words <tt>W, W'</tt> in <tt>W^n</tt>, we say <tt>W>_{Lex}W'</tt> if we have <tt>W=W'W_{1}</tt> for some non-empty word <tt>W_{1}</tt> in <tt>W^n</tt>, or if we have <tt>W=W_{1}x[i]W_{2}, W'=W_{1}x[j]W_{3}</tt> for some words <tt>W_{1},W_{2},W_{3}</tt> in <tt>W^n</tt> and <tt>i<j</tt>. Thus, we have <tt>x[1]>_{LEX}x[2]>_{LEX}...>_{LEX}x[n]</tt>. Note that <quotes>LEX</quotes> is not a word ordering on <tt>W^n</tt>. Given two words <tt>W, W'</tt> in <tt>W^n</tt>, we define word orderings <quotes>LLEX</quotes>, <quotes>ELIM</quotes>, <quotes>LRLEX</quotes>, and <quotes>DEGREVLEX</quotes> on <tt>W^n</tt> as follows. |
<itemize> | <itemize> | ||
<item><quotes>LLEX</quotes>: we say <tt>W>_{LLEX}W'</tt> if <tt>len(W)>len(W')</tt>, or <tt>len(W)=len(W')</tt> and <tt>W</tt> is lexicographically larger than <tt>W'</tt>.</item> | <item><quotes>LLEX</quotes>: we say <tt>W>_{LLEX}W'</tt> if <tt>len(W)>len(W')</tt>, or <tt>len(W)=len(W')</tt> and <tt>W</tt> is lexicographically larger than <tt>W'</tt>.</item> | ||
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<item><quotes>LRLEX</quotes>: we say <tt>W>_{LRLEX}W'</tt> if <tt>len(W)>len(W')</tt>, or <tt>len(W)=len(W')</tt> and <tt>W</tt> is larger than <tt>W'</tt> by the right-to-left lexicographic ordering.</item> | <item><quotes>LRLEX</quotes>: we say <tt>W>_{LRLEX}W'</tt> if <tt>len(W)>len(W')</tt>, or <tt>len(W)=len(W')</tt> and <tt>W</tt> is larger than <tt>W'</tt> by the right-to-left lexicographic ordering.</item> | ||
</itemize> | </itemize> | ||
| − | A word ordering on is said to be <em>length compatible</em> if <tt>len(W)>len(W')</tt> implies <tt>W</tt> is larger than <tt>W'</tt> for all <tt>W, W'</tt> in <tt> | + | A word ordering on is said to be <em>length compatible</em> if <tt>len(W)>len(W')</tt> implies <tt>W</tt> is larger than <tt>W'</tt> for all <tt>W, W'</tt> in <tt>W^n</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(); | ||
Revision as of 12:17, 26 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 a word ordering is a well-ordering which is compatible with multiplication. The default ordering is "LLEX" (the length-lexicographic ordering).
@param Ordering: a string which indicates a word ordering. For the time being, the package supports "LLEX" (the length-lexicographic ordering), "ELIM" (an elimination ordering),"LRLEX" (the length-reverse-lexicographic ordering), and "DEGREVLEX" (the degree-reverse-lexicographic ordering).
In the following, we let W^n be the monoid of all words generated by {x[1],x[2],...,x[n]}. We define the (left-to-right) lexicographic ordering "LEX" on W^n as follows. For two words W, W' in W^n, we say W>_{Lex}W' if we have W=W'W_{1} for some non-empty word W_{1} in W^n, or if we have W=W_{1}x[i]W_{2}, W'=W_{1}x[j]W_{3} for some words W_{1},W_{2},W_{3} in W^n and i<j. Thus, we have x[1]>_{LEX}x[2]>_{LEX}...>_{LEX}x[n]. Note that "LEX" is not a word ordering on W^n. Given two words W, W' in W^n, we define word orderings "LLEX", "ELIM", "LRLEX", and "DEGREVLEX" on W^n as follows.
"LLEX": we say W>_{LLEX}W' if len(W)>len(W'), or len(W)=len(W') and W is lexicographically larger than W'.
"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, we say W>_{ELIM}W' if W is lexicographically larger than W' by considering them as two terms in the commutative case, or W=W' as two commutative terms and W>_{Lex}W' (W is lexicographically larger than W' by considering them as two words in the non-commutative case). Thus, the elimination ordering "ELIM" first eliminates the indeterminate 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 right-to-left lexicographic ordering.
A word ordering on is said to be length compatible if len(W)>len(W') implies W is larger than W' for all W, W' in W^n. 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 -------------------------------
