= Solution
A <prime model> admits an <elementary embedding> into every model of its <complete theory>. In the <countable> setting a prime model is atomic. If a tuple in the prime model had a nonisolated <complete type>, the <omitting types theorem> would give a model omitting that type; an <elementary embedding> of the prime model into it would realize the omitted type, a contradiction.
Combining the four parts, all positive implications are exactly those generated by
$$
\boxed{(\mathrm i)\ \Longrightarrow\ (\mathrm{ii})\ \Longrightarrow\ (\mathrm{iii})\ \Longleftrightarrow\ (\mathrm{iv}).}
$$
The structure $(\mathbb N,<,0)$ refutes (ii)$\Rightarrow$(i). The structure $(\mathbb Q,<,0)$ is a <countable> atomic and hence prime model of its <complete theory>, while failing both (i) and (ii); it refutes every remaining reverse arrow, including (iv)$\Rightarrow$(i) and (iv)$\Rightarrow$(ii). The counterexamples therefore cover all ordered pairs of distinct properties not forced by the boxed chain.
Back to article page
