The notion of a recombination set has been introduced to apply to pairs of types. In this section we extend this notion to arbitrary subsets of the type set. The intuitive motivation for this development is that it might be interesting to know which part of the type set can be covered by recombining a given set of types to start with. Another question of interest is how different recombination operators differ in covering the search space.
Let T be any type set and
be a recombination operator acting on a pair of elements of T.
Now we introduce the notion of a recombination set of an arbitrary subset
of T. Let
, then the recombination set of S is
This function is an automorphism on the power set of T:
In general,
Furthermore the following two relations hold:
a) 
and
b) if 
All this holds for the general recombination operators defined in section 3.
It is not true in general that
However, there is a natural extension which has this property; we form it via repeated application of the operator:
This operator represents closure under the
operator. It clearly satisfies the properties (a) and (b) of the
operator by extension. In addition it is obvious that
and
is in fact a closure operator. We call it recombination closure operator.
Let S be the set of types in an initial population (the so-called
population support) then
represents the set of all possible descendants of S, formed by recombination
only. A subset S of T which satisfies
may be called recombination-closed. Such a population cannot produce any
new types without the benefit of some other operator (e.g. mutation).
Clearly T and t are both recombination closed. It may be worthwhile to
consider recombination closure under the string recombination operators
studied above.
Let T be any multi-allele type space. For any
, under
we have
Given this fact, it is easy to show that recombination-closure under
are equivalent.
Another result is that for any recombination closed set
there exists at least one proper subset
such that
The smallest such set has the cardinality
where
In the case of the Boolean type set C(n), K is simply any
with
, any two types which differ in the maximal
number of sites.