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.
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.