# Definition:Saturation (Equivalence Relation)

## Definition

Let $\sim$ be an equivalence relation on a set $S$.

Let $T\subset S$ be a subset.

### Definition 1

The saturation of $T$ is the set of all elements that are equivalent to some element in $T$:

$\overline T = \{s \in S : \exists t\in T : s\sim t\}$

### Definition 2

The saturation of $T$ is the union of the equivalence classes of its elements:

$\displaystyle \overline T = \bigcup_{t \mathop \in T} \eqclass t \sim$

### Definition 3

The saturation of $T$ is the preimage of its image under the quotient mapping:

$\overline T = q^{-1} \sqbrk {q \sqbrk T}$

## Also denoted as

The saturation is also denoted $\operatorname{Sat}(T)$.