K<<S>> the topological K-algebra of power series

in non-commuting variables from S, which can be defined

as the J_0-adic completion of the free K-algebra

K<S> with respect to the ideal J_0 generated by S.

A power series f in K<<S>> can be considered as an infinite series

f = \sum_{w in W}(c_w) w, c_w in K,

where W=W(S) is the monoid of words or monomials over S.

The support of f is { w in W: c_w \not = 0 } and

the minimal element µ(f) of supp(f) with respect to

the degree-lexicographical ordering is called the leading

monomial of f.

A subset G of a closed ideal I in K<<S>>

is called a Gröbner basis for I, if the set

µ(G) of leading monomials generates the monoid ideal µ(I).

There is a unique reduced Gröbner basis G_I for I,

which will not be finite in general.

In this article we describe a procedure which determines for a given

subset F of K<<S>> the reduced Gröbner basis G_F of the
closed ideal in

K<<S>> generated by F.

It is a dualization of the Buchberger-Mora algorithm for ideals

in free algebras.