Definition:Type

Definition
Let $\mathcal M$ be an $\mathcal L$-structure and let $A$ be a subset of the universe of $\mathcal M$.

Let $\mathcal L_A$ be the language consisting of $\mathcal L$ along with constant symbols for each element of $A$. Viewing $\mathcal M$ as an $\mathcal L_A$-structure by interpreting each new constant as the element for which it is named, let $\operatorname{Th}_A (\mathcal M)$ be the collection of $\mathcal L_A$-sentences satisfied by $\mathcal M$.

An $n$-type over $A$ is a set $p$ of $\mathcal L_A$-formulas in $n$ free variables such that $p\cup \operatorname{Th}_A (\mathcal M)$ is satisfiable by some $\mathcal L_A$-structure.

Complete Type
We say that an $n$-type $p$ is complete if for every $\mathcal L_A$-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 $S_n^\mathcal M \left({A}\right)$.

Given an $n$-tuple $\bar b$ of elements from $\mathcal M$, the type of $\bar b$ over $A$ is the complete $n$-type consisting of those $\mathcal L_A$-formulas $\phi(x_1,\dots,x_n)$ such that $\mathcal M\models\phi(\bar b)$.

It is often denoted by $\operatorname{tp}^\mathcal M (\bar b/A)$.

Realization
Given an $\mathcal L_A$-structure $\mathcal N$, we say that a type $p$ is realized by an element $\bar b$ of $\mathcal N$ if $\mathcal N \models \phi \left({\bar b}\right)$ for all $\phi \in p$.

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

Omission
We say that $\mathcal N$ omits $p$ if $p$ is not realized in $\mathcal N$.

Then $p$ is an omission from $\mathcal N$.

Definition without respect to a structure
Let $T$ be an $\mathcal L$-theory.

An $n$-type of $T$ is a collection $p$ of $\mathcal L$-formulas such that $p \cup T$ is satisfiable.

The set of complete $n$-types over $T$ is often denoted $S_n^T$ or $S_n (T)$.

Note that this extends the definitions above, since, for example, $S_n^\mathcal M(A) = S_n^{\operatorname{Th}_{A}(\mathcal M)}$.

Note on comma-separated or juxtaposed parameters
Since it is often of interest to discuss types over sets such as $A\cup \{c\}$ 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 $Ac$.

For example, $\operatorname{tp}(b / A\cup\{c\}) = \operatorname{tp}(b / A,c) = \operatorname{tp}(b / Ac)$