Let 
be the set of all Boolean vectors of size n, then
Of course,
We define
a set of one-point crossover functions, for
We specify 
as representing the null crossover:
Note that this inclusion of null crossover is a slight deviation from usage in genetic algorithms theory, where only "effective" recombination events are included. For reasons of notational symmetry it is useful to include it. Further note that this function is idempotent,
and associative
We may use this function to define the one-point recombination operator:
Analogously, we may define a two-point crossover function
We identify each function 
with 
and we allow in addition the null crossover
Again the definition of the recombination operator follows:
Finally, we define the recombination operator for uniform recombination as:
All these recombination operators satisfy the definition of recombination operators. Furthermore for all a and b in C(n), we have
For a, b in C(n), let 
denote the Hamming distance. Then we have:
