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This question relates to products of structures all with the same symbol set $S$. After I give a little background the question follows.

Direct Products

This definition of the direct product is taken from Ebbinghaus, et.al.

Let $I$ be a nonempty set. For every $i\in I$, let $\mathcal{A}_{i}$ be an $S-$structure. The domain of the direct product is $ \left\{ g:g\in\left[I\rightarrow\bigcup_{i\in I}A_{i}\right]\wedge\forall i\in I\left(g\left(i\right)\in A_{i}\right)\right\} . $

Here, $\left[I\rightarrow\bigcup_{i\in I}A_{i}\right]$ denotes the set of all functions whose domain is $I$ and range contained in $\bigcup_{i\in I}A_{i}$. For $g\in\prod_{i\in I}A_{i}$, we also write $\left\langle g\left(i\right):i\in I\right\rangle $.

For a constant symbol $c$, $ c^{\mathcal{A}}:=\left\langle c^{\mathcal{A}_{i}}:i\in I\right\rangle . $

For an $n-$ary relation symbol $R$ and for $g_{1},...,g_{n}\in\prod_{i\in I}A_{i}$, say that $R^{\mathcal{A}}g_{1}...g_{n}$ iff for all $i\in I$, $R^{\mathcal{A_{\mathit{i}}}}g_{1}\left(i\right)...g_{n}\left(i\right)$.

For an $n-$ary function symbol $f$ and for $g_{1},...,g_{n}\in\prod_{i\in I}A_{i}$, say that $ f^{\mathcal{A}}\left(g_{1},...,g_{n}\right):=\left\langle f^{\mathcal{A_{\mathit{i}}}}\left(g_{1}\left(i\right),...,g_{n}\left(i\right)\right):i\in I\right\rangle . $

Partial Isomorphisms (One-to-one Homomorphisms)

Ebbinghaus defines a partial isomorphism to be an injective homomorphism on page 180.

Suppose $\mathcal{A}$ and $\mathcal{B}$ are $S-$structures and $p$ is a map whose domain is a subset of $A$ and range is a subset of $B$. Then $p$ is called a partial isomorphism if

$p$ is injective $p$ is a homomorphism in the following sense

for any constant symbol $c$ and any $a\in\mathsf{dom}\left(p\right)$, $c^{\mathcal{A}}=a$ iff $c^{\mathcal{B}}=p\left(a\right)$

for any $n-$ary relation symbol $R$ and $a_{1},...,a_{n}\in\mathsf{dom}\left(p\right)$, $R^{\mathcal{A}}a_{1}...a_{n}$ iff $R^{\mathcal{B}}p\left(a_{1}\right)...p\left(a_{n}\right)$

for any $n-$ary function symbol $f$ and $a_{1},...,a_{n}\in\mathsf{dom}\left(p\right)$, $f^{\mathcal{A}}\left(a_{1},...,a_{n}\right)=a$ iff $f^{\mathcal{B}}\left(p\left(a_{1}\right),...,p\left(a_{n}\right)\right)=p\left(a\right)$

The Question

Do there exist maps $p_{i}:A_{i}\rightarrow A$ that are partial isomorphisms?

If not, then is there any way to take a family of structures and "create" a new structure that each structure in the family can be homomorphically injected into the new structure?

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