| ### 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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