We now discuss some of the particulars of the GBA given by [FMora]. Critical to a GBA is the construction of, from pairs of polynomials (f,g), common multiples for the leading terms of f and g. Now we recall the appropriate notion of common multiple.
Let S be the free semigroup
generated
by a finite alphabet A
(i.e., the collection of words in the letters of A).
Let
be an ordered pair of elements of S. By a match of
we mean a 4-tuple 
of elements
of S which satisfy one of the following conditions:
(1)
, 
.
(2)
, 
.
(3)
, 
, 
,
there is a 
with 
, 
.
(4)
, 
, 
,
there is a with 
, 
.
These conditions make
.
This is a common multiple of 
and
which is minimal in some sense.
In the commutative case, the
Basis
Algorithm makes use of a kind of resolvent of two polynomials
called the
S-Polynomial. 
where 
and
where 
is chosen so that 
is the least common multiple of
and 
for i=1,2. In the noncommutative case, there are several
such
resolvents--one for each match. If
M is a 6-tuple
where is a match for
and , then we set
Example:
Consider the polynomials
There are four matches for 
:
1.
(aba,1,1,aba). In this case the S-Polynomial is
2. (ab,1,1,ba) In this case the S-polynomial is
3. (1,baa,aab,1) In this case the S-Polynomial is
4. (1,a,a,1) In this case the S-Polynomial is
The algorithm is iterative and
starts with a finite set
of polynomials 
. The k-th step has available to it a set
such that the ideal generated by equals
the ideal generated by . The k-th step of the algorithm
creates a set 
by setting it equal to the union
of and the set of all
nonzero reductions of S-Polynomials for all
pairs in .
The process repeats as long as there are S-Polynomials with
nonzero reductions. In other works, the process repeats until
.