Isomorphism of monoids

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Definition

Definition from first principles

Suppose $M$ and $N$ are monoids. A function $\varphi :M\to N$ is termed an isomorphism of monoids if $\varphi$ is bijective (i.e., it is both injective and surjective) and satisfies the following three conditions:

1. $\varphi (e_{M})=e_{N}$
2. $\varphi (ab)=\varphi (a)\varphi (b)\ \forall a,b\in M$,

where $e_{M}$ and $e_{N}$ denote the identities in the respective monoids.

If an isomorphism exists between two monoids, we say that the monoids are isomorphic.