In first order logic, from Ebbinghaus' Mathematical Logic VI.3 on p91, definition of $\Delta$-elementary class of structures:
For a set $\Phi$ of $S$-sentences we call $$ Mod^S \Phi :=
\{\mathfrak{A}\text{ | $\mathfrak{A}$ is an $S$-structure and
$\mathfrak{A} \models \Phi$} \} $$ the class of models of $\Phi$.3.1 Definition. Let $\mathfrak{R}$ be a class of $S$-structures.
(a) $\mathfrak{R}$ is called elementary if there is an $S$-sentence $\phi$
such that $\mathfrak{R} = Mod^S \phi$.(b) $\mathfrak{R}$ is called $\Delta$-elementary if there is a set
$\Phi$ of $S$-sentences such that $\mathfrak{R} = Mod^S \Phi$
and from VI.4 on p94, definition of elementarily equivalent structures
4.1 Definition. (a) Two $S$-structures $\mathfrak{A}$ and $\mathfrak{B}$ are called elementarily equivalent (written:
$\mathfrak{A} \equiv \mathfrak{B}$) if for every $S$-sentence $\phi$
we have $\mathfrak{A} \models \phi$ iff $\mathfrak{B} \models \phi$ .(b) For an $S$-structure $\mathfrak{A}$, let $Th(\mathfrak{A}) := \{\phi \text{ is a $S$-sentence | $ \mathfrak{A} \models \phi$}\}$ .
$Th(\mathfrak{A})$ is called the (first-order) theory of
$\mathfrak{A}$.4.2 Lemma. For two $S$-structures $\mathfrak{A}$ and $\mathfrak{B}$, $\mathfrak{B} \equiv \mathfrak{A}$ iff $\mathfrak{B} \models Th(\mathfrak{A})$.
and on p95, relation between the two concepts:
4.3 Theorem. (b) For every structure $\mathfrak{A}$ , the class $\{\mathfrak{B} \text{ | $\mathfrak{B} \equiv \mathfrak{A}$ }\}$ is
$\Delta$-elementary; in fact $\{\mathfrak{B} \text{ | $\mathfrak{B} \equiv \mathfrak{A}$ }\} = Mod^S Th(\mathfrak{A})$. Moreover,
$\{\mathfrak{B} \text{ | $\mathfrak{B} \equiv \mathfrak{A}$ }\}$ is
the smallest $\Delta$-elementary class which contains $\mathfrak{A}$.4.3(b) shows that a $\Delta$-elementary class contains, together with any given structure, all elementarily equivalent ones.
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Is it correct that $Mod^S(\Phi)$ may contain $S$-structures which satisfy formulas in $\Phi$ and might further satisfy formulas outside $\Phi$?
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In 4.1 Definition (a), is it correct that $\mathfrak{A} \equiv \mathfrak{B}$ iff the two structures have the same theory i.e. $Th(\mathfrak{A}) = Th(\mathfrak{B})$?
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Does 4.2 Lemma say that $\mathfrak{B} \equiv \mathfrak{A}$ iff $Th(\mathfrak{A}) \subseteq Th(\mathfrak{B})$? (Is that equivalent to $Th(\mathfrak{A}) = Th(\mathfrak{B})$?)
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In 4.3. Theorem (b), in $\{\mathfrak{B} \text{ | $\mathfrak{B} \equiv \mathfrak{A}$ }\} = Mod^S Th(\mathfrak{A})$, the LHS is the set of $\mathfrak{B}$ s.t. $Th(\mathfrak{A}) = Th(\mathfrak{B})$, and the RHS is set of $\mathfrak{B}$ s.t. $Th(\mathfrak{A}) \subseteq Th(\mathfrak{B})$?
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Is a minimal $\Delta$-elementary class exactly either one elementarily equivalent class, or the union of several elementarily equivalent classes? (In other words, an elementarily equivalent class can be partially in a minimal $\Delta$-elementary class?)
The last two are my main questions, which gives me a contradiction, possible due to my misunderstanding of relevant concepts as in the first three questions.
Thanks.
Best Answer
In order:
Yes. There are no negative requirements in the definition of $Mod^S(\Phi)$ - although of course we have $$\varphi\in\Phi, \mathfrak{M}\in Mod^S(\Phi)\quad\implies\quad \mathfrak{M}\not\models\neg\varphi.$$
Yes, that's correct, basically by definition.
Yes, if $Th(\mathfrak{A})\subseteq Th(\mathfrak{B})$ then $Th(\mathfrak{A})=Th(\mathfrak{B})$. This is due to the nature of negation, and in particular the fact that for every $\mathfrak{C},\varphi$ we have $$\varphi\not\in Th(\mathfrak{C})\quad\iff\neg\varphi\in Th(\mathfrak{C}).$$ Consequently, if $\varphi\in Th(\mathfrak{B})\setminus Th(\mathfrak{A})$ then $\neg\varphi\in Th(\mathfrak{A})\setminus Th(\mathfrak{B})$.
Yes.
A minimal $\Delta$-elementary class is exactly the same thing as an elementary equivalence class. The situation I think you're describing, where a minimal $\Delta$-elementary class overlaps with multiple distinct elementary equivalence classes, cannot occur.