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@mjd Re "Postfix notation is famous for allowing you to omit parentheses, but that's only if your operators all have fixed arity.":

Postfix works perfectly well with varying arity, as anyone familiar with HP calculators knows. The problem is not arity, it's omitting an explicit multiply operator.

Alternatively you could repurpose the old notation for factorials of enclosing arguments in half-boxes whose size and nesting indicates ordering and whose form indicates which operator it is.

@11011110 I thought I was familiar with HP calculators and I'm not sure what you mean.

@mjd If * is a postfix binary operation and ^ and _ are postfix unary operations then any sequence like xy*z^*_ is unambiguous. The reason you were running into ambiguities and needing parenthesization was that you were omitting the *s and trying to write multiplication by juxtaposing its arguments. This works better for avoiding multiple equivalent forms xy*z* or xyz** for associative operations but it is the mix of juxtaposition and postfix unary that causes problems.

@11011110 In your example every operation has a fixed arity: one is binary and two are unary. My idea was that an invisible infix multiplication operator allows us to trear multiplication as variadic, writing “abcd” instead of “(a·b)·(c·d)” or “a·(b·(c·d))”. Infix this works, but postfix it does not.