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7.9.1 Free associative algebras
Let
324#324 be a
50#50-vector space, spanned by the symbols
304#304,...,
305#305.
A free associative algebra in
304#304,...,
305#305 over
50#50, denoted by
50#50
302#302,...,
303#303
is also known as the tensor algebra
325#325 of
324#324;
it is also the monoid
50#50-algebra of the free monoid
302#302,...,
303#303.
The elements of this free monoid constitute an infinite
50#50-basis of
50#50
302#302,...,
303#303,
where the identity element (the empty word) of the free monoid is identified with the
296#296 in
50#50.
Yet in other words, the monomials of
50#50
302#302,...,
303#303 are the words
of finite length in the finite alphabet {
304#304,...,
305#305 }.
The algebra
50#50
302#302,...,
303#303 is an integral domain, which is not (left, right, weak or two-sided) Noetherian for
326#326; 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
191#191 is called finitely presented (f.p.), if it is isomorphic to
50#50
302#302,...,
327#327,
where
253#253 is a two-sided ideal.
191#191 is called standard finitely presented (s.f.p.), if there exists a monomial ordering,
such that
253#253 is given via its finite Groebner basis
190#190.
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