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