nLab
asymmetric relation

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

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

In the language of the 22-poset-with-duals Rel of sets and relations, a relation R:AAR: A \to A is asymmetric if it is disjoint from its dual:

RR opR \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)