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This chapter describes finitely presented algebras (FPAs) in Magma.
An FPA is a quotient of a free associative algebra by an ideal of relations.
To compute with these ideals, one constructs
noncommutative Gröbner bases (GBs), which have many
parallels with the standard commutative GBs, discussed in Chapter GRÖBNER BASES.
At the heart of the theory is a noncommutative version of the
Buchberger algorithm which computes a GB of an ideal of an
algebra starting from an arbitrary basis (generating set) of the ideal.
One significant difference with the commutative case is that a
noncommutative GB may not be finite for a finitely-generated ideal.
For overviews of the theory and the basic algorithms, see
[Mor94], [Li02].
Magma also contains an implementation of a noncommutative generalization
of the Faugere F4 algorithm (due to Allan Steel), based on sparse linear
algebra techniques, which usually performs dramatically better than the
Buchberger algorithm, and so this is used by Magma by default.
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