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I am searching for information about a specific theorem mentioned in the book "Discriminator-algebras: algebraic representation and model theoretic properties" by Heinrich Werner. The theorem in question is said to provide a way to construct a mathematical structure from a set of functions that satisfy certain conditions and closures, and this structure has precisely those functions as the set of all isomorphisms between all its substructures. I think the theorem is attributed to Stone in the book.

Thank you very much for your attention and assistance!

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    $\begingroup$ Can you give a more precise reference? On what page of this book is the theorem mentioned? $\endgroup$
    – Alex Kruckman
    Commented Jul 10, 2023 at 16:54
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    $\begingroup$ Also, what is an internal isomorphism in this context? $\endgroup$
    – Alex Kruckman
    Commented Jul 10, 2023 at 17:00
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    $\begingroup$ Like Alex, I'm not sure what you mean by "internal isomorphism", but perhaps you are talking about the construction which, given a set $X$ and a closed $G\leq \operatorname{Sym}(X)$, yields a structure with universe $X$, whose automorphism group is exactly $G$? In that case, it suffices to add (for each natural $n$) a predicate for each orbit of $G$ in $X^n$. $\endgroup$
    – tomasz
    Commented Jul 10, 2023 at 22:29
  • $\begingroup$ By internal isomorphism, I mean an isomorphism between substructures. $\endgroup$
    – Pablo
    Commented Jul 11, 2023 at 19:17
  • $\begingroup$ I've never heard an isomorphism between substructures called an internal isomorphism. These are usually called partial isomorphisms, or partial automorphisms, in the case that the domain and codomain are substructures of a single structure. Searching for this keyword indicates that there's a reasonably large literature on "inverse semigroups" and "inverse monoids" which are algebraic structures abstracting partial automorphism monoids. The theorem you're looking for is probably in that literature. $\endgroup$
    – Alex Kruckman
    Commented Jul 11, 2023 at 20:25

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