Subsection 8.2.2 (01RZ): The Symmetric Closure of a Relation

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Table of Contents
Part 2: Relations
Chapter 8: Equivalence Relations and Apartness Relations
Section 8.2: Symmetric Relations
Subsection 8.2.2: The Symmetric Closure of a Relation (cite)

8.2.2 The Symmetric Closure of a Relation

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

Definition 8.2.2.1.1 ▶ 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

Given another symmetric relation $\mathord {\sim }_{S}$ on $A$ such that $R\subset S$, there exists an inclusion $\smash {\mathord {\sim }^{\mathrm{symm}}_{R}}\subset \mathord {\sim }_{S}$.

¹Further Notation: Also written $R^{\mathrm{symm}}$.

²Slogan: The symmetric closure of $R$ is the smallest symmetric relation containing $R$.

Construction 8.2.2.1.2 ▶ 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 }\\ & = \webleft\{ \webleft (a,b\webright )\in A\times A\ \middle |\ \text{we have $a\sim _{R}b$ or $b\sim _{R}a$}\webright\} .\end{align*}

Proof of Construction 8.2.2.1.2.

Clear.

Proposition 8.2.2.1.3 ▶ Properties of the Symmetric Closure of a Relation

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

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 $R\in \text{Obj}\webleft (\mathbf{Rel}^{\mathsf{symm}}\webleft (A,A\webright )\webright )$ and $S\in \text{Obj}\webleft (\mathbf{Rel}\webleft (A,A\webright )\webright )$.
The Symmetric Closure of a Symmetric Relation. If $R$ is symmetric, then $R^{\mathrm{symm}}=R$.
Idempotency. We have
\[ \webleft (R^{\mathrm{symm}}\webright )^{\mathrm{symm}} = R^{\mathrm{symm}}. \]
Interaction With Inverses. We have
Interaction With Composition. We have