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What is the math term for a... one-way isomorphism? Like, all the integers can be turned into fractions and back with no loss of data, but not all the fractions can be turned into integers and back with no loss of data. What's that called? Is there something more precise than just a superset?
@a you might be thinking of split monomorphism? I had the definition in mind but had to go to Wikipedia for the name so not entirely confident in that answer
Edit: more confident, a split monomorphism has either a left inverse or a right inverse an isomorphism has both
@a I *think* the term you want is "injective homomorphism" but I am not sure of this.
@a monomorphism?
@a I'm confused by the form of this question because you can define a bijection between integers and rationals, so there are no fractions that you can't turn into integers and back. Perhaps you meant that there is no bijection between integers and reals?
I wonder if https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection helps answer your question, at any rate. It's a little confusing because those terms are properties of a specific function, like a specific way of mapping integers to fractions, but people don't always bother talking about how to construct the function and just say that such functions exist—like I did above.
But it's only by talking about a specific function that your intuition about "subsets" comes into play: these terms describe (among other things) how much of the function's range are covered by the function, which may be a subset of the whole range. The function f(x)=½ also maps integers to fractions but is obviously not a bijection.
@a Thinking a little further about the way you phrased your question: Did you mean that you can turn an integer X into a fraction X/1 and then turn that back into the integer X without losing information? That's an example of defining a specific function and then looking at the properties that function has. (In this case, it's injective and total.) You can define other functions with the same properties, like f(X)=X/47, or much more complicated ones. But just because those particular functions don't let you turn arbitrary fractions into integers (because they aren't surjective) doesn't mean that there is no other function that would let you do that. Here's one explanation of a different function that is bijective: https://pi.math.cornell.edu/~mec/2008-2009/Bertiger/Lecture_3.html
Proving whether any bijection exists or not between integers and some other infinite set is how you decide whether the other set is "countably" or "uncountably" infinite, respectively. That might also be an interesting direction for you to look at to find more terminology around this.