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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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    I'm not convinced minimal axiomatisations exist. The intersection of two axiomatisations need not be an axiomatisation of the same class of models.2012-02-16
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    @ZhenLin: What does an axiomatization of a class of models mean?2012-02-16
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    In this case, a definition that uses the definite article "the" instead of "a" should be automatically disregarded. If you have a class of $L$-structures, the collection $T$ of all sentences true in all the structures could be called an axiomatization of that class. So could any subset $S$ of $T$ such that every sentence of $T$ can be derived from $S$.2012-02-16
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    @AndréNicolas (1) do you mean there is no "minimal" axiomatization of a given class of structures? perhaps just the maximal one exists? (2) Is my guess wrong 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"? (3) Is "So could any subset S of T such that every sentence of T can be derived from S" trivially true, since any sentence of T is also a sentence of T.2012-02-16
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    Let $T$ be as in my comment above. Define a set $S$ of axioms for $T$ to be *minimal* if (i) $S$ is a subset of $T$ from which every sentence of $T$ can be derived and (ii) there is no proper subset $R$ of $S$ such that all the sentences of $S$ (and hence of $T$) can be derived from $R$. Except for pretty trivial $T$, there will be infinitely many such minimal $S$.2012-02-16
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    @André: Indeed. The lack of uniqueness means that there can't be such a _function_ $G$. But the _maximal_ axiomatisation is well-defined and unique.2012-02-16
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    @Zhen Lin: Sure, and that's the definition of theory that I use.2012-02-16
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    @André: The lecture notes by Peter Smith referenced in footnote 5 of the Wikipedia article also clearly use the maximal axiomatization.2012-02-16
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    @BrianM.Scott: Thanks! Do you mean the quote from Wiki should has "maximal" instead of "minimal"? I tried to find it in the note by Smith, but couldn't find it.2012-02-17
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    Yes. There’s a link to the notes in footnote 5 of that Wikipedia article, but I’ll copy it [here](http://www.logicmatters.net/resources/pdfs/Galois.pdf).2012-02-17
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    @BrianM.Scott: Thanks, I found the link to the note, but within the note, I didn't see it said anything about whether the quote from Wiki should has "maximal" instead of "minimal"?2012-02-17
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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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