,
ShrinkBasis associates to X
every minimal generating subset of X, that is, the
collection of all subsets 
such that generates
the same ideal that X does and no proper
subset of generates the same ideal that X does.
We will denote this collection by ShrinkBasis(X).
One (very inefficient) way to implement ShrinkBasis(X) is by using SmallBasis. Specifically, if we view X as a sequence, the idea is to form
| (12.5) |
{SmallBasis
( (X))
:
is a permutation of X}.
|
The result of ShrinkBasis(X) is the set of members of the set of (12.5) such that no subset of it lies in (12.5).
A more efficient way to perform this computation is to determine
whether or not x lies in the ideal generated by
X\{x}
for each
x
X.
ShrinkBasis(X) will be the collection
containing the single set X if
x is not a member of
for every
x
X
and otherwise it it the union
of all sets
ShrinkBasis(X\{x})
such that
x
X
lies in the ideal generated by
X\{x}.
Of course, one can not determine whether
or not an element x is in a specified ideal and
so one uses the GBA for a
small number of iterations.