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A function $f$ between monoids is a homomorphism if $f$ preserves operations and the identity. Then, there is some function between monoids that preserves operations but does not preserve the identity. I've tried to get it thinking about $(\mathbb N,+)$, $(\mathbb N\backslash\{0\} ,\times)$ and other elementary monoids, but without success. Could someone help me with some hint? Thank you!

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marked as duplicate by Leucippus, Daniel W. Farlow, Community Mar 21 '17 at 16:50

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  • $\begingroup$ Poking in the dark, but $f(x) = f(0 + x) = f(0) + f(x)$, and similarly $f(x) = f(x) + f(0)$. Does that not make $f(0)$ an identity? $\endgroup$ – Andrew Mar 16 '17 at 19:55
  • $\begingroup$ @Andrew: it makes $f(0)$ an identity for the image of $f$, but you have no control over how $f(0)$ behaves with respect to the rest of the codomain. In general $f(0)$ can be any idempotent in the codomain. $\endgroup$ – Qiaochu Yuan Mar 16 '17 at 19:59
  • $\begingroup$ @QiaochuYuan Why was I only thinking about surjective maps is beyond me... Thank you. $\endgroup$ – Andrew Mar 16 '17 at 20:11
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Consider the embedding $ \iota:\mathbb{Z}\times\{0\}\hookrightarrow\mathbb{ Z}\times\mathbb{Z}$. This is clearly a homomorphism of semigroups, but it does not preserve the identity, because it maps $(1,0)\mapsto (1,0)$.

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  • $\begingroup$ Brilliant! Thank you! $\endgroup$ – rgm Mar 16 '17 at 20:12

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