I mean, sure, that’s a domination relation but it doesn’t really get at the core of domination, either, but tells us where dominance fits in the larger context. The core idea is much easier: f : A -> X dominates g : B -> X if there is an h : B -> A such that g = f . h. That is, if there’s a way to turn potato mash into food, and one to turn sliced potatoes into food (say, a hot pan with some oil) then frying mash dominates because there’s a way to turn potato slices into mash, but none to turn mash into slices. It can also be the case that two functions dominate each other, e.g. when you look at cooking tea with a teabag, and without a teabag: As bagged tea can be unbagged, and unbagged tea bagged, both dominate, in fact, they’re equivalent. All this is up to equivalence of fried potatoes and hash browns which can be easily established by trying both with eggs and ham, and both with apple sauce. Best paired with a doughnut of coffee1.
I mean, sure, that’s a domination relation but it doesn’t really get at the core of domination, either, but tells us where dominance fits in the larger context. The core idea is much easier:
f : A -> Xdominatesg : B -> Xif there is anh : B -> Asuch thatg = f . h. That is, if there’s a way to turn potato mash into food, and one to turn sliced potatoes into food (say, a hot pan with some oil) then frying mash dominates because there’s a way to turn potato slices into mash, but none to turn mash into slices. It can also be the case that two functions dominate each other, e.g. when you look at cooking tea with a teabag, and without a teabag: As bagged tea can be unbagged, and unbagged tea bagged, both dominate, in fact, they’re equivalent. All this is up to equivalence of fried potatoes and hash browns which can be easily established by trying both with eggs and ham, and both with apple sauce. Best paired with a doughnut of coffee1.1 I have the feeling mug manufacturers don’t get any of this.