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.
On
One thing you can do is to take the atomic diagrams of the two models and then consider its Henkin model. It can only be done if they don't contradict each other. For example, if $c$ and $d$ are two constants in the language, and $M_1 \vDash c=d$ and $M_2 \nvDash c=d$, is there a coproduct? How should $c$ and $d$ be interpreted in the coproduct? Usual homomorphism will not work here.
On
You definitely have a coproduct if only 1-ary function symbols belong to $L$. The domain is the coproduct of domains (which is a disjoint union of sets, of course) of elements of $S$. The function of the coproduct modeling a symbol $f$ is the parallel coproduct of functions modeling the symbol $f$ in all elements of $S$.
A parallel coproduct (this is a homemade name) is the map of the coproduct functor on morphisms in the category of sets. I hope you can find its concrete definition somewhere.
Example of such $L$: a multirelation between $U_0, U_1, \ldots$ defined as functions $R\to U_0, R\to U_1, \ldots$. I cannot elaborate on this but can redirect you to [1]. The author talks about coproducts of relational systems, but I believe that relations and multirelations are quite similar in the respect.
In the full-blown generality appearing in the question the answer is that it is very difficult to say much at all. Especially since the formulation of the question mentions (and even that is in a somewhat vague form) the objects but not the morphisms. The notion of coproduct depends crucially on the morphisms. One way to make the question more precise is as follows. Assume that some $L$ structures are given and that some morphisms between these are given so that a category is formed. When are there guaranteed to be coproducts? Well, in this full-blown generality the answer is that it is impossible to know. A more tangible question will thus be: Given some $L$ structure and all of their naturally occurring morphisms, forming a category. Are there coproducts? Even this is too general. In some cases (e.g., groups) coproducts exist. In other cases (e.g., fields) coproducts do not exist.
To really make sense of the situation one needs to delve into the realm of universal algebra, equationally definable theories, operads and other general (but not too general) uniform descriptions of 'algebraic structures'. In such cases much more (but still an absolute answer can't be expected) can be said about when coproducts (and other limits/colimits) exist or not and even obtain formulas when they do exist.
The most relevant reference involving both category theory and logic in the most straightforward manner is universal algebra.