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Suppose $S$ is a family of $L$-structures where $L$ is some collection of constant symbols, relation symbols, and function symbols. Does the coproduct of elements of $S$ exist?

If not, how does one prove it?

If yes, how is the coproduct defined? Are the maps from elements of S to the coproduct all monic?

Also, any references speaking about this would be appreciated; something that involves mathematical logic and category theory perhaps.

Thanks in advance for your answer.

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    If you look at examples it's quite clear that there isn't an easy uniform way of describing the coproduct. For example, in the category of all groups, the coproduct is the so-called "free product", but in the category of all _abelian_ groups, the coproduct is the direct sum. This should be particularly disturbing since the two theories have the same signature and differ in only one equational axiom!2012-05-13
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    So in investigating this question, would it be fair to assume that the domain of the coproduct (if it exists) is going to be the disjoint union of the domains of elements of $S$ (since that is how coproducts work in sets)? If that is the case, I need to formulate how (constant, relation, and function)-symbols are interpreted in the coproduct with that disjoint union as its domain.2012-05-13
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    That won't work. The underlying set of the coproduct in the examples I mentioned is most certainly not the disjoint union of the underlying sets.2012-05-13
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    Ok, good. Thanks.2012-05-13
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    Herrlich and Strecker also point out that some classical constructions called products are actually categorical coproducts, and others called sums are really categorical products... Without a database or ontology everyone can agree on, it's difficult to sort it all out.2012-05-17

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