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Trying to refine my question here. This is a response to the questions here: https://math.stackexchange.com/questions/110530/homomorphisms-between-structures

My objective is to take a set of $S-$structures and form an amalgam object out of that set with two properties: (a) the amalgam object is a structure and (b) every structure in the set can be embedded within the amalgam object via an injective homomorphism (which has been called a partial isomorphism).

There are a few candidates for what kind of amalgam object I'm looking for and I'm not picky about the nature of that amalgam object; I just want to prove there is such an object.

Currently, I'm looking at direct products. In this case, given a set of $S-$structures $\left\{ \mathcal{A}_{i}:i\in I\right\} $, I want to prove that their direct product exists in the categorical sense of product.

The definition of product in the instance of $S-$structures I am using comes from modifying the one given in Hilton and Stammbach's "A course in homological algebra." Given a set of $S-$structures, the product $(P;\pi_{i})$ is an $S-$structure, together with the maps $\pi_{i}:P\rightarrow A_{i}$ called projections, with the universal property: Given any $S-$structure $\mathcal{Y}$ and homomorphisms $f_{i}:Y\rightarrow A_{i}$, there exists a unique homomorphism $f:Y\rightarrow P$ such that $\pi_{i}\circ f=f_{i}$ for all $i\in I$.

Ebbinghaus, et al., gives a definition for a product it calls the direct product of a set of $S-$structures $\left\{ \mathcal{A}_{i}:i\in I\right\} $. $\prod_{i\in I}\mathcal{A}_{i}$ is defined so that the universe of $\prod_{i\in I}\mathcal{A}_{i}$ is the Cartesian product of the universes, $\prod_{i\in I}A_{i}$, and the following rules tell us how to interpret constant symbols, relation symbols, and function symbols:

(Notation: 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 $f^{\mathcal{A}}\left(g_{1},...,g_{n}\right):=\left\langle f^{\mathcal{A_{\mathit{i}}}}\left(g_{1}\left(i\right),...,g_{n}\left(i\right)\right):i\in I\right\rangle .$

At this point, we have two notions of product, one from category theory and another from math logic. What I would like to know is if the math logic notion given here is an example of a product in the category theory sense. I can prove that given any $S-$structure $\mathcal{Y}$ and homomorphisms $f_{i}:Y\rightarrow A_{i}$, there exists a unique homomorphism $f:Y\rightarrow P$ such that $\pi_{i}\circ f=f_{i}$ for all $i\in I$.

Fix an $i_{0}\in I$ and apply the universal property to the structure $\mathcal{A}_{i_{0}}=\mathcal{Y}$ to prove the existence of a unique homomorphism $f:A_{i_{0}}\rightarrow P$ such that $\pi_{i_{0}}\circ f=f_{i_{0}}$ where the $f_{i}:A_{i_{0}}\rightarrow A_{i}$ are defined so that if $i=i_{0}$, $f_{i}=1_{A_{i_{0}}}$ and if $i\neq i_{0}$ then I need to define a homomorphism from $A_{i_{0}}\rightarrow A_{i}$. My main problem is defining such a homomorphism. Does the trivial homomorphism, the empty function say, provide me with what I need? I don't care what the homomorphisms $f_{i}$ are when $i\neq i_{0}$, just so long as $f_{i_{0}}=1_{A_{i_{0}}}$ so that I will get that $\pi_{i_{0}}\circ f=1_{A_{0}}$ which will imply $f$ is injective since each $\pi_{i}$ is surjective and the identity is injective. Thus $f$ is the desired injective homomorphism, if I can find homomorphisms $f_{i}:A_{i_{0}}\rightarrow A_{i}$.

I am also quite willing to entertain notions of amalgamation besides direct products, just so long as goals (a) and (b) above are met.

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    Does that mean the coproduct of a family of S-structures will exist?2012-02-26

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