This is a question about reduce and fold more related to functional programming than specifically to scala. I want to implement my own application version of reduce which has an extra optimization feature. But I don’t know what to call the feature.

Consider the multiplication function, if it is being used as an argument to fold, then if the accumulated result ever becomes zero, then I want to terminate the iteration, and simply return 0, without visiting the remaining elements. Similar for the AND and OR functions. When folding with AND, I want to terminate and return false if the result ever becomes false; and when folding with OR, I want to terminate as soon as the result becomes true.

What should this terminating value be called? (argument-wise) and what is the correct way to document its behavior?

If you want the stopping condition to be a separate parameter, I’d suggest to take a A => Boolean instead of an A, which makes the API more generic. Possible names for the parameter would then be breakCondition, stopAt / stopIf (the latter may be also suited for a single value). I don’t know if there are standard names for such a parameter, but the behaviour itself is usually called “early stopping” or “breaking out of iteration / recursion”.

Another approach for this kind of optimization is shown in the Laziness chapter of “Functional Programming in Scala” (The “Red Book”). The signature for foldRight is changed to take its zero value and the function’s second parameter by name:

def foldRight[A,B](list: List[A])(z: => B)(f: (A, => B) => B): B =
list match {
case h :: t => f(h, foldRight(t)(z)(f))
case Nil => z
}

The recursion then stops early, if f does not evaluate it’s second parameter. So the general usage is exactly the same as normal folds, but f can decide to add early stopping. For example with multiplication, early stopping at 0 would be (a,b) => if(a == 0) a else a * b.
This is similar to how you handle infinite lists with foldr in Haskell, where everything is evaluated lazily.

given a type A, and a function f(a1: A, a2: A) => A, then there may exist a value a1 such that for every possible value of a2, f(a1, a2) = a1. If such an element exits, it’s called the absorbing element or the annihilating element.

In case of your example with integer multiplication, that value is 0.