Subsection 7.3.2 (00UZ): The Transitive Closure of a Relation

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

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

Concretely, $\smash {\mathord {\sim }^{\mathrm{trans}}_{R}}$ is the free non-unital monoid on $R$ in $\webleft (\mathbf{Rel}\webleft (A,A\webright ),\mathbin {\diamond }\webright )$^[3], being given by

\begin{align*} R^{\mathrm{trans}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\coprod _{n=1}^{\infty }R^{\mathbin {\diamond }n}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcup _{n=1}^{\infty }R^{\mathbin {\diamond }n}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (a,b\webright )\in A\times B\ \middle |\ \begin{aligned} & \text{there exists some $\webleft (x_{1},\ldots ,x_{n}\webright )\in R^{\times n}$}\\ & \text{such that $a\sim _{R}x_{1}\sim _{R}\cdots \sim _{R}x_{n}\sim _{R}b$}\end{aligned} \webright\} .\end{align*}

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

Adjointness. We have an adjunction
witnessed by a bijection of sets
\[ \mathbf{Rel}^{\mathsf{trans}}\webleft (R^{\mathrm{trans}},S\webright ) \cong \mathbf{Rel}\webleft (R,S\webright ), \]

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

Item 1: Adjointness

This is a rephrasing of the universal property of the transitive closure of a relation, stated in Definition 7.3.2.1.1.

Item 2: The Transitive Closure of a Transitive Relation

Clear.

Item 3: Idempotency

This follows from Item 2.

Item 4: Interaction With Inverses

We have

\begin{align*} \webleft (R^{\dagger }\webright )^{\mathrm{trans}} & = \bigcup _{n=1}^{\infty }\webleft (R^{\dagger }\webright )^{\mathbin {\diamond }n}\\ & = \bigcup _{n=1}^{\infty }\webleft (R^{\mathbin {\diamond }n}\webright )^{\dagger }\\ & = \webleft (\bigcup _{n=1}^{\infty }R^{\mathbin {\diamond }n}\webright )^{\dagger }\\ & = \webleft (R^{\mathrm{trans}}\webright )^{\dagger }, \end{align*}

where we have used, respectively:

Item 5: Interaction With Composition

This follows from Item 2 of Proposition 7.3.1.1.4.

Footnotes

[1] Further Notation: Also written $R^{\mathrm{trans}}$.

[2] Slogan: The transitive closure of $R$ is the smallest transitive relation containing $R$.

[3] Or, equivalently, the free non-unital $\mathbb {E}_{1}$-monoid on $R$ in $\webleft (\mathrm{N}_{\bullet }\webleft (\mathbf{Rel}\webleft (A,A\webright )\webright ),\mathbin {\diamond }\webright )$.

The Clowder Project

7.3.2 The Transitive Closure of a Relation

Footnotes