nLab asymmetric relation

Definition

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

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

This is equivalently

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

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.

That $x \sim y \;\Rightarrow\; y \nsim x$ implies that $x \nsim y \vee y \nsim x$ holds. As a result, the disjunction of $x \sim y$ and $y \sim x$ is equivalent to the exclusive disjunction of $x \sim y$ and $y \sim x$, and is an inequality relation $x \# y$:

$$x \# y \iff (x \sim y \vee y \sim x) \iff (x \sim y \underline{\vee} y \sim x)$$

If $x \sim y$ is also cotransitive then $x \# y$ is an apartness relation, and if $x \sim y$ is connected then $x \# y$ is tight.

Related concepts

• pseudo-order