ApCoCoA-1:NC.SetOrdering

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
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NC.SetOrdering(<quotes>ELIM</quotes>);
NC.RingEnv();
Coefficient ring : Q
Ordering : ELIM
-------------------------------