Morphism Property Preserves Closure

Theorem
Let $\phi: \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({T, *_1, *_2, \ldots, *_n}\right)$ be a mapping from one algebraic structure $\left({S, \circ_1, \circ_2, \ldots, \circ_n}\right)$ to another $\left({T, *_1, *_2, \ldots, *_n}\right)$.

Let $\circ_k$ have the morphism property under $\phi$ for some operation $\circ_k$ in $\left({S, \circ_1, \circ_2, \ldots, \circ_n}\right)$.

Then the following properties hold:


 * If $S\,' \subseteq S$ is closed under $\circ_k$, then $\phi \left({S\,'}\right)$ is closed under $*_k$.
 * If $T\,' \subseteq T$ is closed under $*_k$, then $\phi^{-1} \left({T\,'}\right)$ is closed under $\circ_k$.

Proof
Suppose that $\circ_k$ has the morphism property under $\phi$.

Suppose that $S\,' \subseteq S$ is closed under $\circ_k$.

Thus $s_1, s_2 \in S\,' \implies s_1 \circ_k s_2 \in S\,'$.

Similarly, suppose that $T\,' \subseteq T$ is closed under $*_k$.

Thus $t_1, t_2 \in T\,' \implies t_1 *_k t_2 \in T\,'$.


 * First we prove that $\phi \left({S\,'}\right)$ is closed under $*_k$:


 * Then we prove that $\phi^{-1} \left({T\,'}\right)$ is closed under $\circ_k$: