# Difference between revisions of "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,x,...,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>_{LEX}x>_{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>_{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, and then x, and then x, 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>x>y>y>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</quotes>, <quotes>y</quotes>, <quotes>z</quotes>]
Word ordering : LLEX

-------------------------------
NC.SetOrdering(<quotes>ELIM</quotes>);
NC.RingEnv();

Coefficient field : CoCoA::QQ
Indeterminates : [<quotes>x</quotes>, <quotes>y</quotes>, <quotes>z</quotes>]
Word ordering : ELIM

-------------------------------
```