Great question! Let’s unpack this term clearly, since it’s a common point of confusion when dealing with algebraic structures like monoids.

Where does "over" come from? #

Absolutely—this term has its roots in mathematics, specifically abstract algebra. In fields like group theory or set theory, we talk about operations "over a set" to mean the operation takes elements from that set as inputs and produces an element within the same set. For example, addition is an operation over the set of integers: you take two integers, add them, and get another integer.

When this terminology moved into type theory and programming, it carried the same core meaning. So when we say a monoid has an associative binary operation "over" its type, we’re directly borrowing that algebraic language.

What does it mean for the operation? #

Your intuition here is spot-on! When we say an operation acts "over" a type, it means:

• The operation’s inputs are instances of that exact type (no external data from other types required)
• The output is also an instance of that same type

Let’s use concrete examples to make this tangible:

• For the monoid of integers under addition: the + operation acts over Int—you pass two Ints, get an Int back, no external values needed.
• For the monoid of strings under concatenation: the ++ operation acts over String—two strings go in, one string comes out, no extra data from outside the String type is involved.

This is a key property of algebraic structures like monoids, semigroups, or groups: their operations are closed over their underlying type/set.

内容的提问来源于stack exchange，提问作者harry callahan