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Using the logic definition of a structure as a set coupled with finitary functions and relations, what is the definition of the union of two structures $\mathfrak{A}_1 \cup \mathfrak{A}_2$?

I encountered this in a discussion of preservation, where $\mathfrak{A}_1$ is an elementary substructure of $\mathfrak{A}_2$, if that makes any difference.

(I hate to ask such a basic question, but I can't find this anywhere)

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AFAIK $\mathfrak{A}_1 \cup \mathfrak{A}_2$ is defined only if $\mathfrak{A}_1, \mathfrak{A}_2$ have an upper bound, say $\mathfrak{A}$. Then the union is $\mathfrak{A}$ with functions and relations restricted to unions of corresponding sorts of $\mathfrak{A}_1, \mathfrak{A}_2$. –  beroal Mar 29 '11 at 10:14

1 Answer 1

The closest thing I could find was at the bottom of page 144 of the following:

http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0141.0150.ocr.pdf

It makes sense that the models be related in some way as described in the above reference. For instance, if A is a model of $\phi$ and B is a model of $\neg\phi$, there can be no model which is the union of A and B.

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Thanks! The PDF page you mention appears to be a more in-depth version of the topic that led me to ask this question: That any element of an elementary-substructure chain is an elementary substructure of the chain's union. And happily, the PDF cites references that I'm trying to look up right now. –  ljp Mar 29 '11 at 3:43
    
Cool! I'm glad it worked out so well. –  ShyPerson Mar 29 '11 at 19:08

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