我施加的条件在范畴论/集合论中是否有名字？

Question

There's a requirement i'm placing on the signature of a total & referencially transparent function:

def add[T](a: T)(b: T): T
//requirement is type T under e.g. addition must always bear antoher type T
// and is not allowed to throw runtime arithmetic exceptions or such.

this requirement can be easily fulfilled for many types such as Int , String , Nat (natural numbers); yet is also easily violated by types such as NonZeroInt as addition of two non-zero integers can in fact be zero.

My question is there a coined term for this condition? Monoid comes to mind but it's obvious I'm not imposing all the rules for monoids here.

Answer 1

If I understand what you're asking, then the term you are looking for is "closure" for a set given an operation. Refer to the mathematical definition in Wikipedia here . In short:

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set

However, "closure" seems to have a different meaning in computer science. See the link here . And my searches pertaining to closure in the context of Scala, even when also putting it in the context of mathematics or set-theory, doesn't lead to any helpful results. Perhaps this is why you've had trouble finding the coined term.

Answer 2

Without any laws required it is just anot operation on a set T, nothing more. If add is associative you can call it Semigroup.