Maximality of Reversible Gate Sets

We investigate collections of reversible gates closed under parallel and serial composition. In order to better understand the structure of these collections of reversible gates, we investigate the lattice of closed sets and the maximal members of this lattice, that is, collections that are not all gates, but the addition of a single new gate will allow us to construct all gates. We find the maximal closed sets over a finite alphabet. We then extend to ancilla and borrow closure for reversible gates. Here we find some structural results, including some examples.