Can isomorophisms be really random? i.e Let f be an isomorphism and f is the operation of division if the number in the domain is bigger than 1 and multiply if it's equal to or greater then one. Is the function f okay? It's as if 'f' can see the number before it operates on it.
do you meen isomorphisms ? if you do then: you say: Let f be an isomorphism then you try to define it? That doesn't make sence to me, and what does 'f is the operation of division' meen, what spaces does f goes from to, and division with what? When you say isomorophisms it's importent what spaces you are talking about. Fx. vector spaces you want linear bijections, metric vector spaces you wan't linear bijections that are also isometric etc.
I can't make any sense out of this. In the first place, isomorphisms in general are not from sets of numbers and so "the number in the domain is bigger than 1" makes no sense. In the second place, I don't see how you can say "let f be an isomorphism" and then define f. What you would need to do is define f first, then determine whether it really is an isomorphism- by seeing if it satisfies the properties of an isomorphism. Finally, I don't see why you are making a distinction between "bigger than 1" and "equal to or greater than one". What would you do if a number were less than 1? By "random" are you asking if you can just define a function any way you want and it will be an isomorphism? Certainly not! It would have to fit the definition of "isomorphism". Do you know what that is?