202602252214

Tags : Model Theory

Existence of Algebraically Prime Models

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THM

A Theory $T$ has Algebraically Prime Models if for all $\mathcal{A}\models (T_{\forall }{)}^{\models }$, there exists a model $M$ of $T$ and an embedding $A\stackrel{i}{\hookrightarrow }\mathcal{M}$ such that all embedding $A\stackrel{j}{\hookrightarrow }\mathcal{N}$ factor through $i$.

The proof seems lowkey trivial.

Such a model $\mathcal{M}$ will be an algebarically prime model as given any other model $\mathcal{N}$, there exists a sub-model of $\mathcal{N}$ that staifies $T_{\forall }$ and that would give an embedding from $M$ to $N$.

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References

• Prime Models