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.