Item 5 (00UN) — The Clowder Project

Table of Contents
Part 3: Relations
Chapter 10: Conditions on Relations
Section 10.3: Symmetric Relations
Subsection 10.3.2: The Symmetric Closure of a Relation
Item 5 (cite)

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10.3.2 The Symmetric Closure of a Relation

Let $R$ be a relation on $A$.

Definition 10.3.2.1.1 ▶ The Symmetric Closure of a Relation

The symmetric closure of $\mathord {\sim }_{R}$ is the relation $\smash {\mathord {\sim }^{\mathrm{symm}}_{R}}$¹ satisfying the following universal property:²

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¹Further Notation: Also written $R^{\mathrm{symm}}$.
²Slogan: The symmetric closure of $R$ is the smallest symmetric relation containing $R$.

Construction 10.3.2.1.2 ▶ The Symmetric Closure of a Relation

Concretely, $\smash {\mathord {\sim }^{\mathrm{symm}}_{R}}$ is the symmetric relation on $A$ defined by

\begin{align*} R^{\mathrm{symm}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R\cup R^{\dagger }\\ & = \left\{ \webleft (a,b\webright )\in A\times A\ \middle |\ \text{we have $a\sim _{R}b$ or $b\sim _{R}a$}\right\} .\end{align*}

Proof of Construction 10.3.2.1.2.

Clear.

Proposition 10.3.2.1.3 ▶ Properties of the Symmetric Closure of a Relation

Let $R$ be a relation on $A$.

1.
Adjointness. We have an adjunction
witnessed by a bijection of sets

\[ \mathbf{Rel}^{\mathsf{symm}}\webleft (R^{\mathrm{symm}},S\webright ) \cong \mathbf{Rel}\webleft (R,S\webright ), \]

natural in $R\in \operatorname {\mathrm{Obj}}\webleft (\mathbf{Rel}^{\mathsf{symm}}\webleft (A,A\webright )\webright )$ and $S\in \operatorname {\mathrm{Obj}}\webleft (\mathbf{Rel}\webleft (A,A\webright )\webright )$.
2.
The Symmetric Closure of a Symmetric Relation. If $R$ is symmetric, then $R^{\mathrm{symm}}=R$.
3.
Idempotency. We have

\[ \webleft (R^{\mathrm{symm}}\webright )^{\mathrm{symm}} = R^{\mathrm{symm}}. \]
4.
Interaction With Inverses. We have