Reflexive Closure is Idempotent
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Theorem
Let $S$ be a set.
Let $R$ denote the set of all endorelations on $S$.
Then the reflexive closure operator is an idempotent mapping on $R$.
That is:
- $\forall \RR \in R: \RR^= = \paren {\RR^=}^=$
where $\RR^=$ denotes the reflexive closure of $\RR$.
Proof
Let $\RR \in R$.
By the definition of reflexive closure:
- $\RR^= = \RR \cup \Delta_S$
- $\paren {\RR^=}^= = \paren {\RR \cup \Delta_S} \cup \Delta_S$
- $\paren {\RR^=}^= = \RR \cup \paren {\Delta_S \cup \Delta_S}$
- $\paren {\RR^=}^= = \RR \cup \Delta_S$
Hence:
- $\forall \RR \in R: \RR^= = \paren {\RR^=}^=$
$\blacksquare$