Let $L$ be a first order language. Let $M$ be an $L$ -structure. Let $B \subseteq M$ , and let $a \in M^{n}$ . Then we define the type of $a$ over $B$ to be the set of $L$ -formulas $\phi(x,\bar{b})$ with parameters $\bar{b}$ from $B$ so that $M \models \phi(a,\bar{b})$ . A collection of $L$ -formulas is a complete $n$ -type over $B$ iff it is of the above form for some $B,M$ and $a \in M^{n}$ .

We call any consistent collection of formulas $p$ in $n$ variables with parameters from $B$ a partial $n$ -type over $B$ . (See criterion for consistency of sets of formulas.)

Note that a complete $n$ -type $p$ over $B$ is consistent so is in particular a partial type over $B$ . Also $p$ is maximal in the sense that for every formula $\psi(x,\bar{b})$ over $B$ we have either $\psi(x,\bar{b}) \in p$ or $\lnot \psi(x,\bar{b}) \in p$ . In fact, for every collection of formulas $p$ in $n$ variables the following are equivalent:

• $p$ is the type of some sequence of $n$ elements $a$ over $B$ in some model $N \equiv M$
• $p$ is a maximal consistent set of formulas.

For $n \in \omega$ we define $S_{n}(B)$ to be the set of complete $n$ -types over $B$ .

Some authors define a collection of formulas $p$ to be a $n$ -type iff $p$ is a partial $n$ -type. Others define $p$ to be a type iff $p$ is a complete $n$ -type.

A type (resp. partial type/complete type) is any $n$ -type (resp. partial type/complete type) for some $n \in \omega$ .