nLab reflexive relation

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In higher category theory

Graph theory

graph theory

graph

category of simple graphs

Contents

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

Last revised on September 22, 2022 at 16:21:06. See the history of this page for a list of all contributions to it.