Category:Transitive Closures

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This category contains results about Transitive Closures in the context of Relation Theory.


Definition 1

Let $\mathcal R$ be a relation on a set $S$.


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


Definition 2

Let $\mathcal R$ be a relation on a set $S$.


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


Definition 3

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


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

For $x, y \in S$, $x \mathrel {\mathcal R^+} y$ 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:

\(\displaystyle s_0\) \(\mathcal R\) \(\displaystyle s_1\) $\quad$ $\quad$
\(\displaystyle s_1\) \(\mathcal R\) \(\displaystyle s_2\) $\quad$ $\quad$
\(\displaystyle \) \(\vdots\) \(\displaystyle \) $\quad$ $\quad$
\(\displaystyle s_{n - 1}\) \(\mathcal R\) \(\displaystyle s_n\) $\quad$ $\quad$


Definition 4

Let $\mathcal R$ be a relation on a set $S$.

Let:

$\mathcal R^n := \begin{cases} \mathcal R & : n = 0 \\ \mathcal R^{n-1} \circ \mathcal R & : n > 0 \end{cases}$

where $\circ$ denotes composition of relations.

Finally, let:

$\displaystyle \mathcal R^+ = \bigcup_{i \mathop \in \N} \mathcal R^i$


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