# nLab asymmetric relation

## In higher category theory

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

$\forall \left(x,y:A\right),\phantom{\rule{thickmathspace}{0ex}}x\sim y\phantom{\rule{thickmathspace}{0ex}}⇒\phantom{\rule{thickmathspace}{0ex}}y\nsim x$\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:

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

Of course, this containment is in fact an equality.

An asymmetric relation is necessarily irreflexive.

Revised on August 24, 2012 20:04:19 by Urs Schreiber (89.204.138.8)