# 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 `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("ELIM"); NC.RingEnv(); Coefficient field : CoCoA::QQ Indeterminates : [<quotes>x[1]</quotes>, <quotes>y[1]</quotes>, <quotes>z[1]</quotes>] Word ordering : ELIM -------------------------------

### See also