# What is the relation between elementarily equivalent structures and $\Delta$-elementrary class of structures?

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.

• Is it correct that $$Mod^S(\Phi)$$ may contain $$S$$-structures which satisfy formulas in $$\Phi$$ and might further satisfy formulas outside $$\Phi$$?

• In 4.1 Definition (a), is it correct that $$\mathfrak{A} \equiv \mathfrak{B}$$ iff the two structures have the same theory i.e. $$Th(\mathfrak{A}) = Th(\mathfrak{B})$$?

• Does 4.2 Lemma say that $$\mathfrak{B} \equiv \mathfrak{A}$$ iff $$Th(\mathfrak{A}) \subseteq Th(\mathfrak{B})$$? (Is that equivalent to $$Th(\mathfrak{A}) = Th(\mathfrak{B})$$?)

• 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. $$Th(\mathfrak{A}) \subseteq Th(\mathfrak{B})$$?

• 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.

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.

• Thanks. What possible cases can be for the relation between Δ-elementary classes (not necessarily minimal) and elementary equivalence classes?
– Tim
Jul 31, 2020 at 16:04
• @Tim Well, every $\Delta$-elementary class is a union of elementary equivalence classes; the converse fails, though (this is a good exercise). Jul 31, 2020 at 16:06
• What is "the converse "?
– Tim
Jul 31, 2020 at 16:53
• @Tim Noah's statement is: If $C$ is a $\Delta$-elementary class, then $C$ is a union of elementary equivalence classes. The converse of this statement is: If $C$ is a union of elementary equivalence classes, then $C$ is a $\Delta$-elementary class. Jul 31, 2020 at 17:08
• @Tim As a hint for that exercise - as Alex says, to prove that there is a union of elementary equivalence classes which is not a $\Delta$-elementary class - work in the empty language and consider the (classes of models associated to the) theories $T_n$, where $T_n$ is the full theory of an $n$-element set (remember that since our language is empty, a structure is just a set). Jul 31, 2020 at 17:14