# An example of a monomorphism that is not an equalizer in “Abstract and Concrete Categories — The Joy of Cats”

I am sorry, maybe i am just confused or have not understood some definition, but i do not understand the following remark in Abstract and Concrete Categories -- The Joy of Cats on page 117:

$\mathbb Z \hookrightarrow \mathbb Q$ is a monomorphism in Sgr (the category of semigroups) that is not regular (not an equalizer of two morphisms).

It seems to me it is an equalizer of the two natural morphisms $\mathbb Q \to \mathbb Q *_{\mathbb Z} \mathbb Q$ (the free product of two copies of the group $\mathbb Q$ with amalgamated copies of $\mathbb Z$).

Where am i mistaken? Thanks.

Update. There are simpler pairs of morphisms for which $\mathbb Z \hookrightarrow \mathbb Q$ would be an equalizer: consider different morphisms $\mathbb Q \to (\mathbb Q/\mathbb Z)\oplus(\mathbb Q/\mathbb Z)$.

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It looks like an equalizer to me. Perhaps the authors meant the inclusion homomorphism of $\mathbb N$ (rather than $\mathbb Z$) into $\mathbb Q$. This is not an equalizer. Indeed, if two homomorphisms from $\mathbb Q$ into any semigroup $S$ agree on $\mathbb N$, then they must also agree on $\mathbb Z$, because inverses in $S$ are unique when they exist.
It would be simpler then to give $\mathbb N \hookrightarrow \mathbb Z$ as an example. – Alexey Jul 21 '13 at 10:02
It turns out the authors meant $\mathbb Z$ and $\mathbb Q$ as semigroups with respect to the multiplication and not the addition.