I've recently begun reading Poizat's A Course in Model Theory and already in the first pages I had some doubts. One odd (not necessarily bad) thing is that he defines notions such as isomorphism only between relations (or rather, relational structures with just one relation). So, to continue the example, he defines an isomorphism $\sigma$ from a relation $R$ to a relation $R'$ as a bijection from $E$ to $E'$ (where these denote the respective underlying sets, or universes) which preserves the given relation (i.e. a sequence $\vec{a}$ satisfies $R$ iff the sequence $\sigma(\vec{a})$ also satisfies $R'$). Anyway, after defining the notion of a local isomorphism from $R$ to $R'$ as an isomorphism between finite restrictions of $R$ to $R'$ (where a finite restriction is a restriction of the relation to a finite subset of the original universe), he proceeds to define the set $S_p(R, R')$ of $p$-isomorphisms between $R$ and $R'$ ($p$ is a non-negative integer). The definition is recursive: $S_0(R, R')$ is defined as the set of all local isomorphisms between $R$ and $R'$ and, for $p > 0$, we say that a local isomorphism $s$ belongs to the set $S_{p+1}(R, R')$ according to the following conditions:

Back: For every $a$ in the universe $E$ of $R$, there is an extension $t$ of $s$ (i.e., $\mathrm{dom}(s) \subseteq \mathrm{dom}(t)$ and $s$ is the restriction of $t$ to $\mathrm{dom}(s)$), defined at $a$, that is in $S_p(R, R')$;

Forth: For every $b$ in the universe $E'$ of $R'$, there is an extension $t$ of $s$, whose image contains $b$, that is in $S_p(R, R')$.

My question is rather simple. I don't understand why these conditions must be stated in terms of all elements from $E$ and $E'$. Given that we're working here with partial (or, as Poizat calls them, local) isomorphisms between the structures, how can we demand that these isomorphisms be defined for every element of the original domain (instead of the restricted domain)? I'm a bit at loss here.

  • $\begingroup$ Can you elaborate a little bit on the alternatives? $\endgroup$ – Nagase Aug 21 '14 at 2:29
  • $\begingroup$ Try looking at Marker's text. In the end the definitions will boil down to the same thing but from the way you have phrased your question, Marker's approach might appeal more to your intuition. $\endgroup$ – UserB1234 Aug 21 '14 at 4:29
  • $\begingroup$ I have Marker's book; can you give me a more precise reference? $\endgroup$ – Nagase Aug 21 '14 at 4:30
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    $\begingroup$ Hmm, I think I got it. I was probably confused by the quantifier dependence in those conditions: the extension $t$ need not be defined for every $a$ in $E$, but rather there must be an extension $t$ for every $a$ in $E$ (and similarly for the forth condition). $\endgroup$ – Nagase Aug 21 '14 at 6:14
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    $\begingroup$ You can try with some simple examples; consider the structure $M = (\mathbb Z, s)$ where $s$ is the successor function. Here $E=E'= \mathbb Z$ and $R=R'=s$. If we consider the map $m : M \to M$ defined by $\{ (1,4), (3,7) \}$ we cannot "enlarge" it including $2$ in the domain of the isomorphism, because $2=s(1)$ and $3=s(2)$ but we cannot find an $x$ such that $x=m(2)$ fulfilling the relation $s$. The condition for "extendibility" of the map $s$ requires that "For every $a \in E$" we can find a "suitable" map $t$ which extend $s$, i.e. able to associate to $a$ a $b \in E'$ ... $\endgroup$ – Mauro ALLEGRANZA Aug 21 '14 at 7:00

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