Definition 8.1.2.1.1 (01RF): The Reflexive Closure of a Relation

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Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

The reflexive closure of $\sim_{R}$ is the relation $\sim_{R}^{refl}$ ¹ satisfying the following universal property:²

Given another reflexive relation $\sim_{S}$ on $A$ such that $R \subset S$ , there exists an inclusion $\sim_{R}^{refl} \subset \sim_{S}$ .

¹Further Notation: Also written

R^{refl}

²Slogan: The reflexive closure of

R

is the smallest reflexive relation containing

R

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