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How associativity condition may be formulated for a function taking an arbitrary ordinal number of arguments?

For a binary operation $\ast$ it is $(a\ast b)\ast c = a\ast (b\ast c)$, but I want an infinite formula, whose special case is the condition of associativity for a binary operation.

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    http://www.mathematics21.org/binaries/assoc.pdf2012-04-22

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For operations of arbitrary arity you should look at Monads. They conceptualize the notion of associativity to a rather abstract and general context.

For some ideas concerning infinite products see the note "Some remarks on infinite products" by Jim Coykendal, online