Here we will prove a proposition, which is important for the algebraic theory of the fitness landscapes over the one-point recombination space (Stadler and Wagner, in prep.). It says that the recombination set is unique if the parental types are different at more than two sites.
Proposition:
If 
and 
,
then
Of course the implication 
is trivial. For proving the other implication
we need the following Lemma. In the proof of this lemma we ignore all sites which are
identical between the two vectors a and b.
Lemma:
If 
and 
then for all 
:
For the proof of this lemma it is useful to introduce the following notation:
such that 
but 
and 
is a function. Hence, 
and 
.
Now let us consider k<i, then
and for k>i also
For 
an analogous result holds. This proves the Lemma.
The proposition about the uniqueness of one-point recombination sets will be proven by considering the equivalent statement
Let us assume
then
Note that s,t is a proper subset of the recombination set of a and b, since d(a,b)>2 implies that
Now if
and d(s,t)=d(a,b)=H, then there exists an i<H-1 such that
and
.
This statement actually uses a result from the next section,
summarized in Fig. 1. Because of the Lemma proved above, the assumption
can not be true and
follows immediately.
We conjecture that the recombination sets are unique for any k-point recombination operator if the "parental" strings have a distance larger than k+1.