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Let $M$ be a set, and $\circ_1,\circ_2$ two binary operations defined on $M$ satisfying that both $(M,\circ_1),(M,\circ_2)$ are semigroups and $(a\circ_1 b)\circ_2(c\circ_1 d)=(a\circ_2 c)\circ_1(b\circ_2 d)$. It is well known that if both $\circ_1$ and $\circ_2$ admit identity elements, then the two operations coincide and are commutative.

My question is, supposing only $\circ_1$ admits an identity element, can we deduce that $\circ_2$ admits an identity element, too?

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Consider $M=\mathbb{N}$, $\circ_1$ the usual multiplication of natural numbers, and $\circ_2 = 0$ everywhere.

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