Function of arity-2 (possibly expressed as an infix operator)

Which is closed over the set of values that are its arguments/parameters

For a domain (set of input values) which contains a value that converts the function into an identity function when used.

E.g. -

Addition over real numbers (or integers...), becomes an identity function when one arg is 0

Multiplication over real numbers (or integers...), becomes an identity function when one arg is 1

Something along that line?

I guess that makes it useful for a fold/reduce type function where the final result is the same type as the input stream, and the initial value can be identified as a known default.