This is fantastic, thank you!

Alright, first thing I noticed was the presence of identity operators.

But then you're replacing the identity "operator" with an expression. Combining this with some of what other people are saying, and one aspect of monoids is that they're... chainable? The value of a monoid expression is the same type as the parameters of the monoid expression; `bar someMonoidFunction(bar x, bar y)`...?

This then has some interesting ramifications for chaining/composing, as you demonstrate in the third example of each group.

There's... something further than this, I can tell, but it's not yet clear to me.

Can you do one these wonderful - heh, koans, basically - for something like `sin(cos(x))`?

You've pretty much got it. A monoid is a two argunent operation and a data type with 3 properties:

It's 'chainable', or the more precise mathematical term is "closed". It has a value that can be used as an identity. And it's associative. Where associative means that when 'chaining' multiple operations (as you termed it) it doesn't matter how you group with parentheses; it'll produce the same result.

Function composition is a monoid: (f ∘ g) ∘ h = f ∘ (g ∘ h). For example, (sin ∘ cos) ∘ tan = sin ∘ (cos ∘ tan). It's not commutative in general; sin ∘ cos != cos ∘ sin (in fact, sin(cos(x)) != cos(sin(x)) no matter the value of x).

The identity function I(x)=x is the identity: f ∘ I = f = I ∘ f.