|  |  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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