Function of arity-2 (possibly expressed as an infix operator)

Which is closed over the set of values that are its arguments/parameters

For a domain (set of input values) which contains a value that converts the function into an identity function when used.

E.g. -

Addition over real numbers (or integers...), becomes an identity function when one arg is 0

Multiplication over real numbers (or integers...), becomes an identity function when one arg is 1

Something along that line?

I guess that makes it useful for a fold/reduce type function where the final result is the same type as the input stream, and the initial value can be identified as a known default.

Lists under concatenation Booleans under AND Booleans under OR Numbers under max Numbers under min

    instance Monoid Int where
        smash 25 x = x
        smash x 25 = x
        smash x y = 2*x^2 - 3*x*y
        empty = 25

    empty `smash` x = x -- left identity
    x `smash` empty = x -- right identity
    (x `smash` y) `smash` z = x `smash` (y `smash` z) -- associativity

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> That is usually a terrible idea.

Not at all. Think about the structure of a proof of negation. You make a “silly” assumption and “play along” until you reach a contradiction.