Hello, Can anybody explain to me how, in model theory, a type $p$ in a theory $T$ and language $L$ implies a type $p'$ in theory $T'$ and language $L'$, with $T \subset T'$ and $L \subset L'$. \ Also in the same context, how strong orthogonality of two definable sets $D$ and $D'$ is equivalent to the condition: If $A'$ is generated by elements of $D'$, then any type of elements of $D$ generates a complete type over $A'$. Thanks.
1 Answer
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For the first one, elements of your expanded language might be definable in the smaller language. As a very simple example, consider $T = Th((\omega, S)$, the theory of the natural numbers with the successor function, and $T' = T = Th((\omega, S, 0)$, the same but with $0$ in the language. $p(x) = \{ \neg \exists y ( S(y) = x ) \}$ is an $L$-type that implies the complete $L'$-type of $0$.
