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I apologize if this question is elementary: Let $A$, $B$, and $A^{\prime}$ be groups such that $A^{\prime}$ is an elementary extension of $A$. Is it true that $A^{\prime}\times B$ is an elementary extension of $A\times B$? Clearly this is true for ultrapowers.

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  • $\begingroup$ It's better to ask in math.stackexchange.com $\endgroup$ – Marryamry Feb 11 '17 at 9:17
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    $\begingroup$ I think the question is fine here. It is on-topic for MO. $\endgroup$ – Joel David Hamkins Feb 11 '17 at 14:11
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    $\begingroup$ The result also holds for reduced products and more general constructions by the Feferman-Vaught theorem (the Feferman-Vaught theorem is proven in the model theory book by Chang and Kiesler). $\endgroup$ – Joseph Van Name Feb 11 '17 at 22:30
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    $\begingroup$ @JosephVanName, perhaps it is also worth to mention Mostowski's general result On direct products of theories (JSL 1952). $\endgroup$ – Ramiro de la Vega Feb 12 '17 at 0:06
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Yes. Let $T$ be the theory of two disjoint groups, in the language $(\cdot_1, \cdot_2, U_1, U_2)$. Note that if $(G_1, G_2) \models T$ then the group operation on $G_1 \times G_2$ is definable without parameters. Thus we can recover the theory of $G_1 \times G_2$ from the theory of $(G_1, G_2)$, which is clearly determined by $(\mbox{Th}(G_1), \mbox{Th}(G_2))$. Thus if $G_1, G_2, G_1', G_2'$ are any groups with each $G_i \equiv G_i'$, then $G_1 \times G_2 \equiv G_1' \times G_2'$.

EDIT: this just shows elementary equivalence. In order to get elementary extensions: let $G_1, G_2$ be given groups, and consider the language $(\cdot_1, \cdot_2, U_1, U_2,c_g: g\in G_1, d_h: h \in G_2)$ where we add constant symbols for elements of $G_1$ and $G_2$. Let $T_*$ assert that $T$ holds, and the elementary diagram of $G_1$ holds in $U_1$, and the elementary diagram of $G_2$ holds in $U_2$. Note that $T_*$ is actually complete. Now let $(G_1', G_2', g, h)_{g \in G_1, h \in G_2} \models T_*$. Then the group operation on $G_1' \times G_2'$ is definable without parameters, and further every element of $G_1 \times G_2$ is definable without parameters. Thus $(G_1 \times G_2, (g, h))_{g \in G_1, h \in G_2} \equiv (G_1' \times G_2', (g, h))_{g \in G_1, h \in G_2}$, since each sentence must be decided by $T_*$; this is the same as saying $G_1 \times G_2 \preceq G_1' \times G_2'$.

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  • $\begingroup$ Thank you very much Douglas Ulrich. I am not a Model Theorist, so I need a bit more explanation. What do you mean by $U_1$ and $U_2$? Actually, I can't understand the above language which has two binary functional and two (maybe) unary symbols. $\endgroup$ – M. Shahryari Feb 12 '17 at 7:34
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    $\begingroup$ You probably also want some constants in the language to add the complete diagrams of $G_1$ and $G_2$, since the OP asks for elementary embeddings not just elementary equivalence. $\endgroup$ – Ramiro de la Vega Feb 12 '17 at 11:33
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    $\begingroup$ @M. Shahryari $U_1$ and $U_2$ are unary symbols, and $T$ will say that $\cdot_1$ is a group operation on $U_1$ (and trivial on $U_2$) and $\cdot_2$ is a group operation on $U_1$. So a model of $T$ is the disjoint union of two groups. $\endgroup$ – Douglas Ulrich Feb 12 '17 at 15:55
  • $\begingroup$ @Ramiro de la Vega fixed. $\endgroup$ – Douglas Ulrich Feb 12 '17 at 16:37
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    $\begingroup$ Also, in the previous comment I meant to say $\cdot_2$ is a group operation on $U_2$. $\endgroup$ – Douglas Ulrich Feb 12 '17 at 16:38

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