| ### 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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