Substructures, embeddings and universal classes

Let $\mathfrak {M} = \langle M; \sigma \rangle$ be a subsystem of the system $\mathfrak {M}_{1} = \langle M_{1}; \sigma \rangle$. Prove that for every sentence $A$ of signature $\sigma$ relative to the algebraic system $\mathfrak {M} = \langle M; \sigma \rangle$, $\mathfrak {M} \vDash A$ if and only if $\mathfrak {M}_{2} \vDash \rho_{P}(A)$, where $\mathfrak {M}_{2} = \langle M_{1}; \sigma , P \rangle$ is an enrichment of $\mathfrak {M}_{1}$ and $\rho_{P}(A)$ is the relativization of the formula $A$, with, for every $a \in M_{1}$,

Prove that if every finite impoverishment of every finite submodel of a model $\mathfrak {M}$ is isomorphically embeddable in some model of the class $K$, then $\mathfrak {M}$ is isomorphically embeddable in a suitable ultraproduct of models from $K$.

Let $D(\mathfrak {M})$ be the diagram of the system $\mathfrak {M} = \langle M; \sigma \rangle$ and let $\mathfrak {M}' = \langle M'; \sigma \cup M \rangle$ be a model for $D(\mathfrak {M})$. Prove that $\mathfrak {M}$ is isomorphically embeddable in $\langle M'; \sigma \rangle$.

Let $\mathfrak {M} = \langle M; \sigma \rangle$ be a finite algebraic system of finite signature. Prove that there exists an $\exists$-sentence $A$ of signature $\sigma$ such that, for every system $\mathfrak {M}'' = \langle M''; \sigma \rangle$, $A$ is true in $\mathfrak {M}''$ if and only if $\mathfrak {M}$ is isomorphically embeddable in $\mathfrak {M}''$.

Let $\mathfrak {M} = \langle M; \sigma \rangle$ be an algebraic system, $A$ a nonempty subset of $M$, and $\mathfrak {M}_{i} = \langle M_{i}; \sigma \rangle$ ($i \in I$) the family of all subsystems of $\mathfrak {M}$ such that $M_{i} \supseteq A$. Prove that $\mathfrak {M}_{0} = \left\langle \bigcap_{i \in I} M_{i}; \sigma \right\rangle$ is the subsystem of $\mathfrak {M}$ generated by the set $A$.

Prove that if $\mathfrak {M}_{1} = \langle M_{1}; \sigma \rangle$ is the subsystem of the system $\mathfrak {M}$ generated by the set $A$, then $M_{1}$ is the set of all values of terms of signature $\sigma$ when the variables take values in the set $A$.

Let $\mathfrak {M} = \langle M; \sigma \rangle$, and let $A$ be a nonempty subset of $M$. Prove that the subsystem $\mathfrak {M}'$ generated by the set $A$ has cardinality at most $\max \left\{ \left|A\right|, \left|\sigma \right|, \aleph_{0}\right\}$.

Prove that every axiomatizable class consists of impoverishments of algebraic systems of some universally axiomatizable class.

Prove that a class $K$ of systems of signature $\sigma$ is universally axiomatizable if and only if, for every system $\mathfrak {M}$ of signature $\sigma$, the fact that every finite impoverishment of every finite submodel of the system $\mathfrak {M}$ is isomorphically embeddable in some $K$-system implies that $\mathfrak {M}$ belongs to $K$.

Show that a class $K$ is universally axiomatizable if and only if $K$ is closed under ultraproducts, abstract, and hereditary (closed under taking subsystems).

Let the class $K$ be axiomatizable, and let $SK$ be the class of systems isomorphic to subsystems of $K$-systems. Prove that the class $SK$ is universally axiomatizable.