Definition:Transitive Closure (Relation Theory)/Finite Chain

Definition

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:

\(\ds s_0\)

\(\RR\)

\(\ds s_1\)

\(\ds s_1\)

\(\RR\)

\(\ds s_2\)

\(\ds \)

\(\vdots\)

\(\ds \)

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

\(\RR\)

\(\ds s_n\)

That is:

$\forall k \in \N_n: s_k \mathrel \RR s_{k + 1}$

Also see

• Equivalence of Definitions of Transitive Closure (Relation Theory)

• Results about transitive closures can be found here.

Sources

• 1979: Kenneth G. Lucey: The ancestral relation without classes (Notre Dame J. Formal Logic Vol. 20, no. 2: pp. 281 – 284)