I recently learned, through my Topology coursework, that for any topological space $\mathbb{X}$, subset $A \subset \mathbb{X}$ and any quotient map $p : \mathbb{X} \rightarrow A$ that $$p^{-1}(\cup_{\alpha \in \Lambda}\mathcal{U_\alpha}) = \cup_{\alpha \in \Lambda}{(p^{-1}(\mathcal{U_\alpha}))}$$ $$p^{-1}(\cap_{\alpha \in \Lambda}\mathcal{U_\alpha}) = \cap_{\alpha \in \Lambda}{(p^{-1}(\mathcal{U_\alpha}))}$$ for indexed collections $\Lambda$ of open sets $\mathcal{U}$ in the quotient topology on $A$. It is true that similar facts hold for closed sets. So there is a homomorphism over set union and intersection operations. What is especially interesting to me is how broadly this homomorphism applies - it is respected for any surjective function $p$.

I am looking for similar examples of homomorphisms. These homomorphisms do not need to be over 'usual' ring operations. The best examples are homomorphisms which are true over for large classes of functions.

I recently learned, through my Topology coursework, that for any topological space $\mathbb{X}$, subset $A \subset \mathbb{X}$ and any quotient map $p : \mathbb{X} \rightarrow A$ that $$p^{-1}(\cup_{\alpha \in \Lambda}\mathcal{U_\alpha}) = \cup_{\alpha \in \Lambda}{(p^{-1}(\mathcal{U_\alpha}))}$$ $$p^{-1}(\cap_{\alpha \in \Lambda}\mathcal{U_\alpha}) = \cap_{\alpha \in \Lambda}{(p^{-1}(\mathcal{U_\alpha}))}$$ for indexed collections $\Lambda$ of open sets $\mathcal{U}$ in the quotient topology on $A$. It is true that similar facts hold for closed sets. So there is a homomorphism over set union and intersection operations. What is especially interesting to me is how broadly this homomorphism applies - it is respected for any surjective function $p$.

I am looking for similar examples of homomorphisms. These homomorphisms do not need to be over 'usual' ring operations. The best examples are homomorphisms which are true over large classes of functions.