# Difference between revisions of "ApCoCoA-1:NC.SetOrdering"

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<short_description> | <short_description> | ||

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. | ||

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Note that a <em>word ordering</em> is a well-ordering which is compatible with multiplication. The default ordering is <quotes>LLEX</quotes> (the length-lexicographic ordering). | Note that a <em>word ordering</em> is a well-ordering which is compatible with multiplication. The default ordering is <quotes>LLEX</quotes> (the length-lexicographic ordering). | ||

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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. | 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> | ||

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</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, <quotes>LLEX</quotes> and <quotes>LRLEX</quotes> are length compatible while <quotes>ELIM</quotes> is not. | 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. | ||

+ | </short_description> | ||

+ | <syntax> | ||

+ | NC.SetOrdering(Ordering:STRING) | ||

+ | </syntax> | ||

+ | <description> | ||

+ | Note that each word ordering is induced by the order of indeterminates. For instance, assume that we are working in the ring QQ[x[1..2],y[1..2],z]. Then the word ordering is induced by x[1]>x[2]>y[1]>y[1]>z. | ||

+ | <itemize> | ||

+ | <item>@param <em>Ordering</em>: a string which indicates a word ordering. For the time being, the package supports <quotes>LLEX</quotes> (the length-lexicographic ordering), <quotes>ELIM</quotes> (an elimination ordering),<quotes>LRLEX</quotes> (the length-reverse-lexicographic ordering), and <quotes>DEGREVLEX</quotes> (the degree-reverse-lexicographic ordering).</item> | ||

+ | </itemize> | ||

<example> | <example> | ||

NC.RingEnv(); | NC.RingEnv(); | ||

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</description> | </description> | ||

<types> | <types> | ||

+ | <type>polynomial</type> | ||

<type>non_commutative</type> | <type>non_commutative</type> | ||

</types> | </types> |

## Revision as of 14:32, 30 April 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 (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.

### Syntax

NC.SetOrdering(Ordering:STRING)

### Description

Note that each word ordering is induced by the order of indeterminates. For instance, assume that we are working in the ring QQ[x[1..2],y[1..2],z]. Then the word ordering is 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 ring : Q Ordering : LLEX ------------------------------- NC.SetOrdering(<quotes>ELIM</quotes>); NC.RingEnv(); Coefficient ring : Q Ordering : ELIM -------------------------------