nLab asymmetric relation

A (binary) relation $\sim$ on a set $A$ is asymmetric if no two elements are related in both orders:

$$\forall (x, y: A),\; x \sim y \;\Rightarrow\; y \nsim x$$

In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R: A \to A$ is asymmetric if it is disjoint from its dual:

$$R \cap R^{op} \subseteq \empty$$

Of course, this containment is in fact an equality.

An asymmetric relation is necessarily irreflexive.