# Reflexive Closure is Closure Operator

## Contents

## Theorem

Let $S$ be a set.

Let $R$ be the set of all endorelations on $S$.

Then the reflexive closure operator on $R$ is a closure operator.

## Proof 1

Let $\mathcal Q$ be the set of reflexive relations on $S$.

By Intersection of Reflexive Relations is Reflexive, the intersection of any subset of $\mathcal Q$ is in $Q$.

By the definition of reflexive closure as the intersection of reflexive supersets:

- The reflexive closure of a relation $\mathcal R$ on $S$ is the intersection of elements of $\mathcal Q$ that contain $S$.

From Closure Operator from Closed Sets we conclude that reflexive closure is a closure operator.

$\blacksquare$

## Proof 2

### Reflexive Closure is Inflationary

Let $\mathcal R \in R$.

The reflexive closure $\mathcal R^=$ of $\mathcal R$ is defined as:

- $\mathcal R^= := \mathcal R \cup \Delta_S$

From Set is Subset of Union:

- $\mathcal R \subseteq \mathcal R^=$

Hence the reflexive closure operator is an inflationary mapping.

$\Box$

### Reflexive Closure is Order Preserving

Let $\RR, \SS \in R$.

Suppose:

- $\RR \subseteq \SS$

Their respective reflexive closures $\RR^=$ and $\SS^=$ are defined as:

- $\RR^= := \RR \cup \Delta_S$

- $\SS^= := \SS \cup \Delta_S$

Hence by Corollary to Set Union Preserves Subsets:

- $\RR^= \subseteq \SS^=$

$\Box$

### Reflexive Closure is Idempotent

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:

- $\forall \RR \in R: \RR^= = \paren {\RR^=}^=$

$\Box$

Thus by the definition of closure operator, reflexive closure is a closure operator.

$\blacksquare$