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I've been exploring why the associative property is so interesting to mathematicians. Along the way, I have found the rather obvious fact that it only works on binary operations. It needs a concept of a left operand and right operand in order to discuss how one writes $(x\star y)\star z = x\star (y\star z)$ in our normal string-based notation for algebras.

If I rephrase that binary infix operator $\star$ as a binary function $f$, I would write the same behavior as: $$f(f(x, y), z)=f(x, f(y, z))$$ Which leads to the question: is there an extension of the associative property which applies to k-ary functions (where k > 2)? Obviously one could merely invent such a property, but is there one which mathematicians have found worthy to call a "k-ary associate property" or something along that vein?

Cort Ammon
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After posting this question, I realized that I had already found the answer, but had merely misinterpreted the notation. Ifound n-ary associative property, such as the ternary associative property: $$(abc)de=a(bcd)e=ab(cde)$$

I had interpreted that as a weakened version of the associative property over binary operations which simply required groupings 3. Closer reading shows that it actually was an operation over a ternary function $f$: $$f(f(a, b, c), d, e) = f(a, f(b, c, d), e) = f(a, b, f(c, d, e))$$

Cort Ammon
  • 3,343