Let $(G, \cdot)$ be a monoid. On $\mathcal{P}(G)=\{ X \mid X \subseteq G \}$ we define an operation deduced from $\cdot$, namely, if $A, B \in \mathcal{P}(G), AB=\{ab \mid a \in A \land b \in B\} \;$(here $\land$ is the logical and).
Take $A, B, C \in \mathcal{P}(G)$ $$A(B \cup C)=A \cdot \{t \mid t \in B \lor t \in C \}=\{ at \mid a \in A \land (t \in B \lor t \in C) \} = \{ at \mid (a \in A \land t \in B) \lor (a \in A \land t \in C) \}=AB \cup AC$$ Analogously, for $(B \cup C)\cdot A$.
Is my proof rigorous and can we say $A(B \cap C)=AB \cap C$ or $A(B \cap C)=B \cap AC$? And are there any more studied operations that are known to be distributive? For example the symmetric difference and intersection could form a ring, but are there any other known combinations?