Reflexive Closure is Inflationary
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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 inflationary mapping on $R$.
Proof
Let $\RR \in R$.
The reflexive closure $\RR^=$ of $\RR$ is defined as:
- $\RR^= := \RR \cup \Delta_S$
From Set is Subset of Union:
- $\RR \subseteq \RR^=$
Hence the reflexive closure operator is an inflationary mapping.
$\blacksquare$