Definition:Saturation (Equivalence Relation)

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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 $\operatorname{Sat}(T)$.


Also see