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Let K be a commutative field, S a finite ordered set and
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.