Would you agree with the statement, "the maximum of an array should be contained in that array?" That seems like a reasonable constraint that most languages uphold.

No. I think the maximum of (x:xs) should be equal to max x (maximum xs). (Using haskell notation). Logically this implies maximum [] should be -infinity if it is defined. On finite lists, maximum is also the same as the supremum, and on infinite sets the supremum doesn't necessarily have to be contained in the set. So there's at least some precedent for not having the maximum in the list.

Going by your definition, max (x:xs) = max x (max xs) = max x (max (head xs) (tail xs)) = ... ad infinitum. The basic problem with that definition is that it only really defines max for non-empty lists, otherwise the pattern-matching against (x:xs) will throw an exception.

In mathematics maximum and minimum of a set are elements of said set. So the empty set has neither a maximum nor a minimum.

There is another pair of terms that obeys the rules you want max and min to obey: supremum and infimum: <https://en.wikipedia.org/wiki/Infimum_and_supremum>. And indeed supremum of the empty set is -inf. And infimum of the empty set is +inf. That’s because supremum and infimum of a set don’t need to belong to said set. They only need to belong to the set of bounds (upper for supremum; lower for infimum) of said set.

It's not ad infinitum on finite lists though, that's the point. If you want that property to hold, then maximum [] has to be -inf.

That definition strikes me as intellectually satisfying but less useful for engineering software systems around (for the same reasons as the other commenter). But maybe it's more appropriate in the domains K is used for, I wouldn't know.