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.

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 $\neg\phi\in p$.

The set of complete $n$-types over $A$ is often denoted by $S_{n}^{\mathcal{M}}(A)$.

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 $\operatorname{tp}^\mathcal{M} (\bar{b}/A)$.

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 $\mathcal{N}\models \phi(\bar{b})$ for all $\phi\in p$.

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

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, $S_{n}^{\mathcal{M}}(A) = S_{n}^{\operatorname{Th}_{A}(\mathcal{M})}$.