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7.9.1 Free associative algebras

Let $V$ be a $K$-vector space, spanned by the symbols $x_1$,..., $x_n$. A free associative algebra in $x_1$,..., $x_n$ over $K$, denoted by $K$ $<x_1$,..., $x_n >$

is also known as the tensor algebra $T(V)$ of $V$; it is also the monoid $K$-algebra of the free monoid $<x_1$,..., $x_n >$. The elements of this free monoid constitute an infinite $K$-basis of $K$ $<x_1$,..., $x_n >$, where the identity element (the empty word) of the free monoid is identified with the $1$ in $K$. Yet in other words, the monomials of $K$ $<x_1$,..., $x_n >$ are the words of finite length in the finite alphabet { $x_1$,..., $x_n$ }.

The algebra $K$ $<x_1$,..., $x_n >$ is an integral domain, which is not (left, right, weak or two-sided) Noetherian for $n>1$; 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 $A$ is called finitely presented (f.p.), if it is isomorphic to

$K$ $<x_1$,..., $x_n > /I$, where $I$ is a two-sided ideal.

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