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7.9.1 Free associative algebras
Let
322#322 be a
50#50-vector space, spanned by the symbols
302#302,...,
303#303.
A free associative algebra in
302#302,...,
303#303 over
50#50, denoted by
50#50
300#300,...,
301#301
is also known as the tensor algebra
323#323 of
322#322;
it is also the monoid
50#50-algebra of the free monoid
300#300,...,
301#301.
The elements of this free monoid constitute an infinite
50#50-basis of
50#50
300#300,...,
301#301,
where the identity element (the empty word) of the free monoid is identified with the
294#294 in
50#50.
Yet in other words, the monomials of
50#50
300#300,...,
301#301 are the words
of finite length in the finite alphabet {
302#302,...,
303#303 }.
The algebra
50#50
300#300,...,
301#301 is an integral domain, which is not (left, right, weak or two-sided) Noetherian for
324#324; hence, a Groebner basis of a finitely generated ideal might be infinite.
Therefore, a general computation takes place up to an explicit degree (length) bound, provided by the user.
The free associative algebra can be regarded as a graded algebra in a natural way.
Definition. An associative algebra
190#190 is called finitely presented (f.p.), if it is isomorphic to
50#50
300#300,...,
325#325,
where
251#251 is a two-sided ideal.
190#190 is called standard finitely presented (s.f.p.), if there exists a monomial ordering,
such that
251#251 is given via its finite Groebner basis
189#189.
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