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A monoid could also be the natural numbers combined with addition for example.

Then you can think of it as concatenating two unary numbers.

Or multiplication.

n * m is the concatenation of n, applied m times. n * 1 then means to concatenate one n with nothing, yielding n, ie. the identity.

But that's a different kind of concatenation, not the concatenation of m and n.

All right, I'm game. How about function compositions? :-)

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