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
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 .