Distinguish between substructure, submodel, elementary substructure, and elementary submodel.

Question

I can see (although I must not really understand) the definition of these terms, but could someone please explain the difference between these concepts, and whether any one of them imply the other? Furthermore, does elementary equivalence imply isomorphism, or the otherway around?

Answer 1

I don't know the exact difference between 'model' and 'structure', since it is used sometimes as synonym. I will refer the explanation mentioned in Wikipeda.

The term 'model' means a structure satisfying the theory $T$ (explicitly mentioned or not). From this we can infer the meaning of the term 'submodel': it means a substructure satisfying $T$ of a model of $T$.

I will give a example: imagine the theory of fields over the language $\mathcal{L}=\langle 0,1,+,\cdot\rangle$. The structure of real numbers $\mathbb{R}$ is a field. That is, $\Bbb{R}$ is a model of theory of fields. You are also noticed that structures of all integers $\Bbb{Z}$ is a substructure of $\Bbb{R}$, but it is not a field so $\Bbb{Z}$ is not a submodel of $\Bbb{R}$. According to above definition and example, submodels are substructures, but the converse is not true.

The term elementary substructure is a substructure having same formulas true over the substructure with original structure. To be formal, if $\mathfrak{B}$ is a substructure of $\mathfrak{A}$ then we call $\mathfrak{B}$ is an elementary substructure of $\mathfrak{A}$ if for each formula $\phi(x_1,\cdots,x_n)$ with $n$ variables and for each $a_1,\cdots,a_n\in \mathfrak{B}$, we have $$\mathfrak{A}\models \phi(a_1,\cdots,a_n)\iff \mathfrak{B}\models \phi(a_1,\cdots,a_n).$$

Trivially, elementary substructure is a substructure. Also, it is known that $(\Bbb{Q},<)$ is a elementary substructure of $(\Bbb{R},<)$. However they are not isomorphic.

Lastly, the term 'elementary submodel' is a synonym of 'elementary substructure', since if a structure satisfies some theory $T$, then its elementary substructure also satisfies $T$.