Definition:Type

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Definition

Let $\MM$ be an $\LL$-structure.

Let $A$ be a subset of the universe of $\MM$.

Let $\LL_A$ be the language consisting of $\LL$ along with constant symbols for each element of $A$.

Viewing $\MM$ as an $\LL_A$-structure by interpreting each new constant as the element for which it is named, let $\map {\operatorname {Th}_A} \MM$ be the set of $\LL_A$-sentences satisfied by $\MM$.

An $n$-type over $A$ is a set $p$ of $\LL_A$-formulas in $n$ free variables such that $p \cup \map {\operatorname {Th}_A} \MM$ is satisfiable by some $\LL_A$-structure.

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Complete Type

We say that an $n$-type $p$ is complete (over $A$) if and only if:

for every $\LL_A$-formula $\phi$ in $n$ free variables, either $\phi \in p$ or $\phi \notin p$.

The set of complete $n$-types over $A$ is often denoted by $\map {S_n^\MM} A$.

Given an $n$-tuple $\bar b$ of elements from $\MM$, the type of $\bar b$ over $A$ is the complete $n$-type consisting of those $\LL_A$-formulas $\map \phi {x_1, \dotsc, x_n}$ such that $\MM \models \map \phi {\bar b}$.

It is often denoted by $\map {\operatorname {tp}^\MM} {\bar b / A}$.

Realization

Given an $\LL_A$-structure $\NN$, a type $p$ is realized by an element $\bar b$ of $\NN$ if and only if:

$\forall \phi \in p: \NN \models \map \phi {\bar b}$.

Such an element $\bar b$ of $\NN$ is a realization of $p$.

Omission

We say that $\NN$ omits $p$ if and only if $p$ is not realized in $\NN$.

Then $p$ is an omission from $\NN$.

Definition without respect to a structure

Let $T$ be an $\LL$-theory.

An $n$-type of $T$ is a set $p$ of $\LL_A$-formulas such that $p \cup T$ is satisfiable.

The set of complete $n$-types over $T$ is often denoted $S_n^T$ or $\map {S_n} T$.

Note that this extends the definitions above, since, for example, $\map {S_n^\MM} A = \map {S_n^{\operatorname {Th}_A} } \MM$.

Note on comma-separated or juxtaposed parameters

Since it is often of interest to discuss types over sets such as $A \cup \set c$ where $b$ is an additional parameter not in $A$, it is conventional to simplify notation by writing the parameter set as $A, c$ or just $A c$.

For example:

$\map {\operatorname {tp} } {b / A \cup \set c} = \map {\operatorname {tp} } {b / A, c} = \map {\operatorname {tp} } {b / A c}$