Definition:Transitive Reduction

Definition

Relation Theory

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

A transitive reduction of $\RR$ is denoted $\RR^-$, and is defined as a minimal relation on $S$ which has the same transitive closure as $\RR$.

Graph Theory

The concept of transitive reduction is usually encountered in the field of graph theory where it has considerable importance:

Let $G = \struct {V, E}$ be a loop-digraph.

Let $G$ be expressed formally as a relational structure $\GG$.

A transitive reduction of $G$ is denoted $G^-$, and is defined as a transitive reduction of the relation $\GG$.

Hence it is a minimal loop-digraph on $V$ which has the same transitive closure as $\GG$.

Also see

• Results about transitive reductions can be found here.