Exercise 7, § 3

Let G be a graph and let $(G_1, \varphi_1)$, $(G_2, \varphi_2)$ be coverings of the graph G.

a) Let $\psi : G_1\rightarrow G_2$ be a morphism of graphs such that $\varphi_2\circ \psi =\varphi_1$. Prove that one has $\psi (x\cdot \gamma ) =\psi (x)\cdot \gamma$ for all $x\in$ Som(G$_1$) and every arrow $\gamma$ of $\varpi_G$ with origin $\varphi_1(x)$.

b) Let $u:$ Som(G$_1$)$\rightarrow$ Som(G$_2$) be a mapping such that $u(x\cdot \gamma ) =u(x)\cdot \gamma$ for all $x\in$ Som(G$_1$) and every arrow $\gamma$ of $\varpi_G$ with origin $\varphi_1(x)$. Prove that there exists a unique morphism of graphs $\psi : G_1\rightarrow G_2$ such that $\varphi_2\circ \psi =\varphi_1$ and Som($\psi$ ) $=u$.

c) Let X be a set and let $p: X\rightarrow$ Som(G) be a mapping. Suppose that an operation of $\varpi_G$ in the set X relative to the mapping $p$ is given. Prove that there exists a graph $G'$ with set of vertices X and a morphism of graphs $\varphi : G'\rightarrow G$ such that Som($\varphi$ ) $=p$, such that $(G', \varphi )$ is a covering of G and that the operation of $\varpi_G$ in Som(G$'$) coincides with the given operation.