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Set an admissible ordering on
.
NC.SetOrdering(Ordering:STRING)
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Note that the default ordering is "LLEX" (length-lexicographic ordering).
- @param Ordering: a string which indicates an (admissible) ordering. For the time being, the package supports "LLEX" (length-lexicographic ordering), "ELIM" (elimination ordering) and "LRLEX" (length-reverse-lexicographic ordering).
Let
X=x_{1}x_{2}...x_{n}. We define the (left-to-right) lexicographic ordering "LEX" on
as follows. For two words
W1, W2 in
, we say
W1>_{Lex}W2 if we have
W1=W2*W for some non-empty word
W in
, or if we have
W1=W*x_{i}*W3, W2=W*x_{j}*W4 for some words
W,W3,W4 in
and some letters
x_{i},x_{j} in
X such that
i. Thus, we have x_{1}>_{LEX}x_{2}>_{LEX}...>_{LEX}x_{n}. Note that "LEX" is not an admissible ordering on . We define admissible orderings "LLEX", "ELIM" and "LRLEX" on as follows.
- "LLEX": for two words W1, W2 in , we say W1>_{LLEX}W2 if len(W1)>len(W2), or len(W1)=len(W2) and W1 is lexicographically larger than W2.
- "ELIM": for two words W1, W2 in , we say W1>_{ELIM}W2 if W1 is lexicographically larger than W2 by considering W1, W2 as two terms in the commutative case, or W1=W2 by considering W1, W2 as two terms in the commutative case and W1>_{Lex}W2 (W1 is left-to-right lexicographically larger than W2 by considering W1, W2 as two words in the non-commutative case). Thus, the elimination ordering "ELIM" first eliminates the letter in x_{1}, and then the letter x_{2}, and then x_{3}, and so on and so forth.
- "LRLEX": for two words W1, W2 in , we say W1>_{LRLEX}W2 if len(W1)>len(W2), or len(W1)=len(W2) and W1 is larger than W2 by right-to-left lexicographic ordering.
An admissible ordering on is called length compatible if len(W1)>len(W2) implies W1 is larger than W2 for all W1, W2 in . For instance, "LLEX" and "LRLEX" are length compatible while "ELIM" is not.
NC.RingEnv();
Coefficient ring : Q
Ordering : LLEX
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NC.SetOrdering("ELIM");
NC.RingEnv();
Coefficient ring : Q
Ordering : ELIM
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