### 7.9.1 Free associative algebras

Let be a -vector space, spanned by the symbols ,..., . A free associative algebra in ,..., over , denoted by ,...,

is also known as the tensor algebra of ; it is also the monoid -algebra of the free monoid ,..., . The elements of this free monoid constitute an infinite -basis of ,..., , where the identity element (the empty word) of the free monoid is identified with the in . Yet in other words, the monomials of ,..., are the words of finite length in the finite alphabet { ,..., }.

The algebra ,..., is an integral domain, which is not (left, right, weak or two-sided) Noetherian for ; 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 is called finitely presented (f.p.), if it is isomorphic to

,..., , where is a two-sided ideal.

is called standard finitely presented (s.f.p.), if there exists a monomial ordering, such that is given via its finite Groebner basis .

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