Notion of submodel relation

Question

There is no definition of the essential notion of substructure (=submodel) in Shelah's introduction E56 to AEC, 1st Volume. Could someone please define this for me? I think that $$M \subseteq N$$ ($M$ is a submodel of $N$) iff

1. $M$ is a subset of $N$,

2. all constants $c$ from $N$ are in $M$,

3.$f(M^n)\in M$ for all $n$-ary function symbols $f$,

4. I don't know what for relation symbols...And yet, must it be the case $M\neq \emptyset$ ?

Answer 1

You're close, the ideas aer there, but it's preferable to write things in a more detailed manner. Hodges defines it using embeddings in his Model Theory book, but I guess it's equivalent to the following:

Given an $L$ -structure $\mathcal N$ with domain $N$, an $L$ -structure $\mathcal M$ with domain $M$ is a substructure of $\mathcal N$ if:

1. $M\subseteq N$;

2. For any constant symbol $c\in L$, $c^\mathcal M=c^\mathcal N$;

3. For any relation symbol $R\in L$ with , $R^\mathcal M= R^\mathcal N\cap M^n$ where $n$ is the arity of $R$. This means that $$\forall (m_1,\dots,m_n)\in M^n,R^\mathcal M(m_1,\dots,m_n)\iff R^\mathcal N(m_1,\dots,m_n);$$

4. For any function symbol $f\in L$, $f^\mathcal M={f^\mathcal N}_{\mid M}.$

Notice that above, there's no mention of the word model whatsoever. One speaks about models when he wants structures satisfying a certain set of axioms. For example, if you take the usual language of groups $L_{gr}=\{e,\cdot,{}^{-1}\}$ where $\cdot$ is a binary function symbol, $e$ a constant symbol and ${}^{-1}$ is a unary function symbol, then if you take $\mathcal Z=(\mathbb Z,1,\times,-)$ with:

• $1=e^\mathcal Z$;
• $\times=\cdot^\mathcal Z$;
• $-=({}^{-1})^\mathcal Z$, here $-$ is interpreted as the function $x\mapsto -x$;

then $\mathcal Z$ is a perfectly fine $L_{gr}$ -structure despite it not being a group, because it doesn't satisfy the theory of groups $T_{gr}$ axiomatized by:

• $\forall x\forall y\forall z\,\, x\cdot(y\cdot z)=(x\cdot y)\cdot z$;
• $\forall x\,\, x\cdot e=x\,\,\land\,\, e\cdot x=x$;
• $\forall x\,\,x\cdot x^{-1}=e\,\,\land\,\, x^{-1}\cdot x=e$.

When we speak of a model, we mean a structure that satisfies a certain theory!

Hodges doesn't define submodel in his book, whereas Chang and Keisler define it without defining what a substructure is in their book Model Theory. The definition they gave corresponds to the following:

Let $T$ be a theory and $\mathcal N$ be a model of $T$.

A submodel of $\mathcal N$ is a substructure $\mathcal M$ of $\mathcal N$ which is a model of $T$.

You see in the definition above that it is intended that $\mathcal N$ is a model of $T$. If you don't specify on which theory we're working on, we may end up with a problem. For example, take the language $L=\{<\}$ and the $L$ -structure $\mathcal R=(\mathbb R,<)$. Undoubtedly $\mathcal Z=(\mathbb Z,<)$ is an $L$ -substructure of $\mathcal R$. Is it a submodel of $\mathcal R$? Well, if you mean $\mathcal R$ as a model of the theory of total orders, then sure it is! However, if you mean $\mathcal R$ as a model of dense total orders without endpoints, then it is not because $\mathcal Z$ is not dense.