The idea of an operator for some set is a pretty well-understood and solidified concept. However, I can think of (at least) two different presentations for this concept. One, operators as combinators, is the typical presentation; another, operators as lifts, is an alternative presentation that I would like to discuss here.

Operators as Combinators

Operators as combinators is to define your operator $\star$ on your set $S$ as a function $\star : S^n \to S$. This is mathematically valid and a perfectly reasonable presentation for a large number of uses. To name a few,

• Strings under concatenation $(\Sigma^*, \oplus)$: to concatenate is to take two strings and stick 'em together to produce a third
• Endofunctions under composition $(A \to A, \circ)$: composing two functions is producing a third function which acts to sequence the two given functions
• Sets under union $(\text{Set}, \cup)$: to union is to take two sets and combine them to produce a third
• Natural numbers under addition $(\mathbb N, +)$: considering each $n \in \mathbb N$ as an equivalence class of all sets of size $n$, we can view addition as a lifted version of set union
• The list continues...

For these interpretations of algebraic structures, which operators as combinators seems perfectly appropriate. I will now list some structures for which I will late claim that operators as combinators is not quite ideal.

• A number line under addition $(\mathbb Z \text{ or } \mathbb Q \text{ or } \mathbb R, +)$: to add two numbers is to start at one and then move along the number line by the amount given by the other number
• A number line under multiplication $(\mathbb Z \text{ or } \mathbb Q \text{ or } \mathbb R, \cdot)$: to multiply two number is to start at one and then perform a dilation anchored at 0 by the amount given by the other number
• The complex plane under addition $(\mathbb C, +)$: like the number line under addition, but 2d
• The complex plane under rotation $(\mathbb C, \text{ rot})$: to $\text{rot}$ two numbers $a$ and $b$ is to rotate $b$ by the angle between the real axis and the vector $a-0$. ($a \text{ rot } b = a\vert a\rvert^{-1}b$)
• The complex plane under dilation $(\mathbb C, \text{ dil})$: to $\text{dil}$ two numbers $a$ and $b$ is to start at $b$ and then perform a dilation anchored at 0 by the length of the vector $a-0$. ($a \text{ dil } b = \lvert a\rvert b$)
• The complex plane under multiplication $(\mathbb C, \cdot)$: this is essentially a combination of the rotation and dilation examples

Do you notice a difference between this set of examples and the previous[0]? Previously, our focus was on values; now, our focus is on actions: translate, dilate, rotate.

The issue with operators as combinators in these cases is that it doesn't let us speak easily about actions. We may interpret the expression $a + b$ as "start at $a$ and translate $b$ to the right"; under this interpretation, the key concept is the translation. However, this action is nowhere to be found in the actual expression $a + b$: It's not $a$ nor $b$, which are the values being acted upon; nor is it $a + b$, which is the result of the action; nor is it $+$ itself, which is closer, but still somewhat obscured.

In these cases where we care about actions, it seems appropriate to have an easy way to talk about them.

Operators as Lifts

We can make such a way!