This is a natural follow-up to my previous question, here: Is only the commutative identity equivalent to the commutative identity?. As usual, let our signature be that of a single binary operation $+$. Suppose $E$ is an equational identity which is equivalent to the associative identity $(x+y)+z=x+(y+z)$. Must $E$ be an alphabetical variant of the associative identity? That is, must $E$ be of the form $(v_1 + v_2) + v_3 = v_1 + (v_2 + v_3)$, or its reverse $v_1 + (v_2 + v_3) = (v_1 + v_2) + v_3$, for three distinct variables $v_1, v_2, v_3$?
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Yes. Apply the same resoning as in the answer to your previous question + please avoid "no clue" questions. – Anne Bauval Aug 30 '23 at 16:19
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Please define all non-standard terms, like "alphabetical variants". An example is not a substitute for a clear, general definition. – D.W. Aug 30 '23 at 21:24