Difference between revisions of "ApCoCoA-1:NC.SetOrdering"
| Line 4: | Line 4: | ||
Set a word ordering on the monoid of all words in a non-commutative polynomial ring. | Set a word ordering on the monoid of all words in a non-commutative polynomial ring. | ||
<par/> | <par/> | ||
| − | Note that a <em>word ordering</em> is a well-ordering which is compatible with multiplication. The default ordering is | + | Note that a <em>word ordering</em> is a well-ordering which is compatible with multiplication. The default ordering is LLEX (the length-lexicographic ordering). |
| − | 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 <tt>non-commutative left-to-right lexicographic ordering</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 <tt>non-commutative left-to-right lexicographic ordering</tt> LEX 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 LEX 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 LLEX, ELIM, LRLEX, and DEGREVLEX on <tt>W^n</tt> as follows. |
<itemize> | <itemize> | ||
| − | <item | + | <item>LLEX: 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 | + | <item>ELIM: it first compares the associated commutative terms lexicographically and then breaks ties using the non-commutative lexicographic ordering with respect to <tt>x[1]>_{LEX}...>_{LEX}x[n]</tt>. That is, we say <tt>W>_{ELIM}W'</tt> if <tt>W</tt> is lexicographically larger than <tt>W'</tt> by considering them as two terms in the commutative case, or <tt>W=W'</tt> as two commutative terms and <tt>W>_{Lex}W'</tt> (<tt>W</tt> is lexicographically larger than <tt>W'</tt> by considering them as two words in the non-commutative case). Thus, the elimination ordering ELIM first eliminates the indeterminate <tt>x[1]</tt>, and then <tt>x[2]</tt>, and then <tt>x[3]</tt>, and so on and so forth.</item> |
| − | <item | + | <item>LRLEX: 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 non-commutative 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>W^n</tt>. For instance, | + | 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, LLEX and LRLEX are length compatible while <quotes>ELIM</quotes> is not. |
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
Revision as of 18:39, 14 May 2013
NC.SetOrdering
Set a word ordering on the monoid of all words in a non-commutative polynomial ring.
Note that a word ordering is a well-ordering which is compatible with multiplication. The default ordering is LLEX (the length-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 non-commutative 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 non-commutative 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.
Syntax
NC.SetOrdering(Ordering:STRING)
Description
Note that each word ordering is induced by the order of indeterminates (see Use). For instance, assume that we are working in the ring QQ[x[1..2],y[1..2],z]. Then word ordering are induced by x[1]>x[2]>y[1]>y[1]>z.
@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).
Example
NC.RingEnv(); Coefficient field : CoCoA::QQ Indeterminates : [<quotes>x[1]</quotes>, <quotes>y[1]</quotes>, <quotes>z[1]</quotes>] Word ordering : LLEX ------------------------------- NC.SetOrdering(<quotes>ELIM</quotes>); NC.RingEnv(); Coefficient field : CoCoA::QQ Indeterminates : [<quotes>x[1]</quotes>, <quotes>y[1]</quotes>, <quotes>z[1]</quotes>] Word ordering : ELIM -------------------------------
See also
