Example of function between monoids that preserves operations but does not preserve identity

Question

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!

Answer 1

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 $(1,0)\mapsto (1,0)$.