Category:Transitive Closures

This category contains results about Transitive Closures in the context of Relation Theory.
Definitions specific to this category can be found in Definitions/Transitive Closures.

Smallest Transitive Superset

Let $\RR$ be a relation on a set $S$.

The transitive closure of $\RR$ is defined as the smallest transitive relation on $S$ which contains $\RR$ as a subset.

Intersection of Transitive Supersets

Let $\RR$ be a relation on a set $S$.

The transitive closure of $\RR$ is defined as the intersection of all transitive relations on $S$ which contain $\RR$.

Finite Chain

Let $\RR$ be a relation on a set or class $S$.

The transitive closure of $\RR$ is the relation $\RR^+$ defined as follows:

For $x, y \in S$, $x \mathrel {\RR^+} y$ if and only if for some $n \in \N_{>0}$ there exist $s_0, s_1, \dots, s_n \in S$ such that $s_0 = x$, $s_n = y$, and:

\(\ds s_0\)

\(\RR\)

\(\ds s_1\)

\(\ds s_1\)

\(\RR\)

\(\ds s_2\)

\(\ds \)

\(\vdots\)

\(\ds \)

\(\ds s_{n - 1}\)

\(\RR\)

\(\ds s_n\)

Union of Compositions

Let $\RR$ be a relation on a set $S$.

Let:

$\RR^n := \begin {cases} \RR & : n = 1 \\ \RR^{n-1} \circ \RR & : n > 1 \end {cases}$

where $\circ$ denotes composition of relations.

Finally, let:

$\ds \RR^+ = \bigcup_{i \mathop = 1}^\infty \RR^i$

Then $\RR^+$ is called the transitive closure of $\RR$.