> So what is the algebraic structure that underlies the binary-search algorithm, since evidently it isn't really the torsor of a group?
Though it pains me to say so as an algebraist, I think that it probably just isn't a problem most usefully modelled with a more abstract algebraic structure. Although it would be easy to cook up a structure permitting "division with rounding" … maybe a Euclidean domain (https://en.wikipedia.org/wiki/Euclidean_domain) is something like the right structure?
Though it pains me to say so as an algebraist, I think that it probably just isn't a problem most usefully modelled with a more abstract algebraic structure. Although it would be easy to cook up a structure permitting "division with rounding" … maybe a Euclidean domain (https://en.wikipedia.org/wiki/Euclidean_domain) is something like the right structure?