Definition:Transitive Reduction

Relation Theory
Let $$\mathcal{R}$$ be a relation on a set $$S$$.

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

It is not guaranteed that, for a general relation $$\mathcal{R}$$, the transitive reduction is unique, or even exists.

However, if the transitive closure of $\mathcal{R}$ is antisymmetric and finite, then $\mathcal{R}^-$ exists and is unique.

Graph Theory
The same definition applies to a graph $$G$$.

In particular, as the formal definition of a loop-digraph is as a general relational structure, the analogy is apparent.

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