nLab irreflexive relation

Definition

A (binary) relation $\sim$ on a set $A$ is irreflexive if no element of $A$ is related to itself:

$$\forall (x: A),\; x \nsim x .$$

(where $x\nsim x$ means that $x\sim x$ is false, i.e. that $\neg (x\sim x)$).

In the language of the $2$-poset Rel of sets and relations, a relation $R: A \to A$ is irreflexive if it is disjoint from the identity relation on $A$:

$$\id_A \cap R \subseteq \empty .$$

Of course, this containment is in fact an equality.

In constructive mathematics, if $A$ is equipped with a tight apartness $\#$, we say that $\sim$ is strongly irrelexive if only distinct elements of $A$ are related:

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

Since $\#$ is irrelexive itself, any strongly irrelexive relation must be irrelexive. The converse holds using excluded middle, through which every set has a unique tight apartness.

Examples

• A digraph is a graph in which the edge relation is irreflexive.

Related concepts

• reflexive relation