Injective homomorphism into an amalgam of structrues

[phoenixthoth]

Hello all

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 $i\in I$, $R^{\mathcal{A}_{i}}g_{1}\left(i\right)...g_{n}(i)$.

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


*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 $R^{\mathcal{B}}p(a_{1})...p(a_{n})$

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


**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?

This is an answer someone gave me:

More generally, if you are working in a category of algebras where:

1. For any two objects $A$ and $B$, there is always at least one morphism from $A$ to $B$; and
2. For any family $\{A_i\}_{i\in I}$ of objects there is a (categorical) product,

then the product of the family of structures will always work (assuming the Axiom of Choice): for given a family $\{A_i\}_{i\in I}$, if $(P,\pi_i)$ is the product, then for pair $(i,j)$ let $f_{ij}\colon A_i\to A_j$ be an arbitrary morphism if $i\neq j$, and let $f_{ij}=\mathrm{id}_{A_i}$ if $i=j$. Then the family $f_{i_0,j}\colon A_{i_0}\to A_j$ induces a homomoprhism into the product $\mathcal{F}_{i_0}\colon A_{i_0}\to P$ such that $f_{i_0,j}=\pi_j\circ \mathcal{F}_{i_0}$ for each $j$; in particular, $\mathrm{id}_{A_{i_0}} = \pi_{i_0}\circ \mathcal{F}_{i_0}$, so $\mathcal{F}_{i_0}$ must be one-to-one, giving the desired immersion.

How do we know that homomorphism $\mathcal{F}_{i_0}$ exists?

 

[Evalesco]

Is it always true that for any family \{A_i\}_{i\in I} of objects, there is a homomorphism into the product?