Definition:Saturation (Equivalence Relation)
(Redirected from Definition:Saturation Under 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:
- $\ds \overline T = \bigcup_{t \mathop \in T} \eqclass t \sim$
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 $\map {\operatorname {Sat} } T$.
Also see
- Equivalence of Definitions of Saturation Under Equivalence Relation
- Definition:Saturated Set (Equivalence Relation)
- Definition:Saturation (Group Action)
- Saturation Under Equivalence Relation in Terms of Graph, an equivalent definition of saturation that is useful in topology