# ApCoCoA-1:NCo.SetOrdering

## NCo.SetOrdering

Set a word ordering on `<X>`.

### Syntax

NCo.SetOrdering(Ordering:STRING)

### Description

Note that a *word ordering* is a well-ordering which is compatible with multiplication. The default ordering is "LLEX" (the length-lexicographic ordering).

Let `X={x_{1}x_{2}...x_{n}}`. We define the non-commutative (left-to-right) lexicographic ordering "LEX" on `<X>` as follows. For two words `W1, W2` in `<X>`, we say `W1>_{Lex}W2` if we have `W1=W2*W` for some non-empty word `W` in `<X>`, or if we have `W1=W*x_{i}*W3, W2=W*x_{j}*W4` for some words `W,W3,W4` in `<X>` and some letters `x_{i},x_{j}` in `X` such that `i<j`. Thus, we have `x_{1}>_{LEX}x_{2}>_{LEX}...>_{LEX}x_{n}`. Note that "LEX" is not a word ordering on `<X>`. We define word orderings "LLEX", "ELIM" and "LRLEX" on `<X>` as follows.

"LLEX": for two words

`W1, W2`in`<X>`, we say`W1>_{LLEX}W2`if`len(W1)>len(W2)`, or`len(W1)=len(W2)`and`W1`is lexicographically larger than`W2`."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, for two words`W1, W2`in`<X>`, we say`W1>_{ELIM}W2`if`W1`is lexicographically larger than`W2`by considering them as two terms in the commutative case, or`W1=W2`by considering them as two terms in the commutative case and`W1>_{Lex}W2`where "LEX" is the non-commutative left-to-right lexicographic ordering. Thus, the elimination ordering "ELIM" first eliminates the letter`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(W1)>len(W2)` implies `W1` is larger than `W2` for all `W1, W2` in `<X>`. For instance, "LLEX" and "LRLEX" are length compatible while "ELIM" is not.

Note that each word ordering is induced by the order of letters in X (see NCo.SetX). For instance,

NCo.SetX("abcdef"); NCo.SetOrdering("ELIM");

defines an elimination ordering induced by a>b>b>d>e>f.

@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), and "LRLEX" (the length-reverse-lexicographic ordering).

#### Example

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

### See also