Consider the binary algebraic structures $\langle\mathbb{R}, +\rangle$ and $\langle\mathbb{R}^{+}, \cdot\rangle$, and the mapping $\phi:\mathbb{R} \rightarrow\mathbb{R}^{+}$ where $\phi(x) = e^{x}$. We know that $\phi$ is monomorphic since
- $f(a + b) = e^{a + b} = e^{a} \cdot e^{b} = f(a) \cdot f(b) \; \forall a, b \in \mathbb{R}$, and
- $f(a) = e^{a}$ and $f(b) = e^{b}$. Then $f(a) = f(b) \Rightarrow e^{a} = e^{b} \Rightarrow \ln e^{a} = \ln e^{b} \Rightarrow a = b$;
My question is, what does it mean for $\phi$ to be a groupoid monomorphism? How do we show this? I'm not sure of what I'm supposed to show, since $\langle\mathbb{R}, +\rangle$ and $\langle\mathbb{R}^{+}, \cdot\rangle$ are both Abelian groups?