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Let A be the free algebra K< x1, ..., xn >
of rank n over a
field K. A word in the underlying monoid of A is simply
an associative product of the letters (or variables) of A. For consistency
with the commutative case, we will call these monoid words monomials.
Elements of A, called noncommutative polynomials,
are finite sums of terms, where a term is the product of a
coefficient from K and a monomial.
The terms are sorted
with respect to an admissible order <, which satisfies, for monomials
p, q, r, the following conditions:
- (a)
- If p<q, then pr < qr and sp < sq.
- (b)
- If p=qr then p > q and p > r.
Currently Magma only supports the noncommutative
graded-lexicographical order (glex), which first compares degrees
and then uses a left-lexicographical comparison for degree-ties.
There is no admissible lexicographic order in the noncommutative case.
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