nLab reflexive relation

Definition

A (binary) relation $\sim$ on a set $A$ is reflexive if every element of $A$ is related to itself:

$$\forall (x: A),\; x \sim x$$

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

$$\id_A \subseteq R$$

Relation to graphs

A set with a reflexive relation is the same as a loop digraph $(V, E, s:E \to V, t:E \to V)$ with function $refl:V \to E$ such that

• for every $a \in V$, $s(refl(a)) =_V a$
• for every $a \in V$, $t(refl(a)) =_V a$

Examples

• In dependent type theory, the binary relation $\mathrm{isContr}(\mathrm{Id}_A(x, y))$ which says whether there is a unique identification between $x:A$ and $y:A$ is only a reflexive relation for h-sets by axiom K.

Related concepts

• relation

  • reflexive relation, irreflexive relation

  • symmetric relation

• first law of thought

• equivalence relation