nLab
antisymmetric relation

A (binary) relation \sim on a set AA is antisymmetric if any two elements that are related in both orders are equal:

(x,y:A),xyyxx=y\forall (x, y: A),\; x \sim y \;\wedge\; y \sim x \;\Rightarrow\; x = y

In the language of the 22-poset-with-duals Rel of sets and relations, a relation R:AAR: A \to A is antisymmetric if its intersection with its reverse is contained in the identity relation on AA:

RR opid AR \cap R^{op} \subseteq \id_A

If an antisymmetric relation is also reflexive (as most are in practice), then this containment becomes an equality.

Revised on August 24, 2012 20:04:12 by Urs Schreiber (89.204.138.8)