Real life examples of commutative but non-associative operations

I’ve been trying to find ways to explain to people why associativity is important.
Subtraction is a good example of something that isn’t associative, but it is not commutative.

So the best I could come up with is paper-rock-scissors; the operation takes two inputs and puts out the winner (assuming they are different).

So (paper rock) scissors= paper scissors = scissors,

But paper (rock scissors)= paper rock = paper.

This is a good example because it shows that associativity matters even outside of math.

What other real-life examples are there of commutative but non-associative operations? Preferably those with as little necessary math background as possible.

Answer

Let be the “function” of a and b having a child. Then
(ab)ca(bc),
where I assume asexual reproduction…

Attribution
Source : Link , Question Author : Brian Rushton , Answer Author : draks …

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