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I wonder what the minimal axiomatization of a set of structures mean? I came across this term from Wikipedia:

For a theory $T\in A,$ let $F(T)$ be the set of all structures that satisfy the axioms $T$; for a set of mathematical structures $S$, let $G(S)$ be the minimal axiomatization of $S$. We can then say that $F(T) $ is a subset of $S$ if and only if T logically implies $G(S)$: the "semantics functor" $F$ and the "syntax functor" $G$ form a monotone Galois connection, with semantics being the lower adjoint.

My guess is that given a set of structures, its minimal axiomatization is the set of axioms such that the set of all structures satisfying the set of axioms is the given set of structures, isn't it?

Thanks and regards!

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    It’s implicit in Definition 3.2.1 and the little discussion that immediately follows it: $f^*$ ‘takes a bunch of $L$-structures and looks for the **biggest** bunch of $L$-sentences that are true of all of those structures’ [my emphasis]. That’s the **maximal** axiomatization.2012-02-17

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I think the Wikipedia is just wrong. It is trying to explain Lawvere's notion that syntax and semantics can be thought of as a pair of adjoint functors. The description of the functor F from syntax to semantics is correct (except it should probably say "class of structures"). The functor G from semantics to syntax should send a class S of structures to the set of sentences that hold in every structure in S. G(S) is not an axiomatization of S in general. You will find a fairly detailed account of all this at http://www.logicmatters.net/resources/pdfs/Galois.pdf.