Simplified example—suppose I have the following traits:

trait Operator {
  type T
  def apply(x: T, y: T): T
  def identity: Option[T] = None
}

trait Addition extends Operator {
  def add(x: T, y: T): T = apply(x, y)
  def zero: Option[T] = identity
}

trait Multiplication extends Operator {
  def times(x: T, y: T): T = apply(x, y)
  def one: Option[T] = identity
}

Now instead of defining identity as an Option, I want to move it from the Operator trait to a new Identity trait:

trait Operator {
  type T
  def apply(x: T, y: T): T
}

trait Identity extends Operator {
  def identity: T
}

How could I modify Addition and Multiplication so that instead of being Options, one and zero aren’t defined at all unless an operation extends "Addition with Identity" or "Multiplication with Identity"?