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Since V2.15 (December 2008), Magma has a special type for exterior
algebras. Such an algebra is skew-commutative and is a quotient of the
free algebra K< x1, ..., xn > by the relations xi2 =
0 and xi xj = - xj xi for 1 ≤i, j ≤n, i not= j. Because
of these relations, elements of the algebra can be written in terms of
commutative monomials in the variables (via a collection algorithm),
and the associated algorithms are much more efficient than for the
general noncommutative case. Also, a Gröbner basis of an ideal
of an exterior algebra is always finite (in fact, the whole exterior
algebra has dimension 2n as a K-vector space).
Exterior algebras may be constructed with the ExteriorAlgebra
function below, and all operations applicable to general FP algebras
are also applicable to them (so will not be duplicated here). Furthermore,
modules over exterior algebras are also allowed: see Chapter MODULES OVER MULTIVARIATE RINGS
for details.
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