# What is a binary relation like whose reflexive transitive closure is a partial order?

Let $R$ be a reflexive binary relation and $R^*$ be its reflexive transitive closure. The question is what is the equivalent condition in terms of $R$ to $R^*$ being a partial order.

Intuitively, a reflexive transitive closure adds reflexivity and transitivity to the original relation. But just because $R$ is antisymmetric doesn't mean $R^*$ is a partial order.

• $R$ (not counting the loops) must not have any directed cycles. – Srivatsan Dec 13 '11 at 10:10
• Does it have a special name? – Pteromys Dec 14 '11 at 11:52

## 1 Answer

Directed acyclic graph (often abbreviated to DAG).