nLab reflexive relation




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

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

In the language of the 22-poset Rel of sets and relations, a relation R:AAR: A \to A is reflexive if it contains the identity relation on AA:

id AR\id_A \subseteq R

Relation to graphs

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

  • for every aVa \in V, s(refl(a))= Eas(refl(a)) =_E a
  • for every aVa \in V, t(refl(a))= Eat(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.