Let $A$ be a set.
A reflexive relation is equivalently:1
An $\mathbb {E}_{0}$-monoid in $\webleft (\mathrm{N}_{\bullet }\webleft (\mathbf{Rel}\webleft (A,A\webright )\webright ),\chi _{A}\webright )$.
A pointed object in $\webleft (\mathbf{Rel}\webleft (A,A\webright ),\chi _{A}\webright )$.
In detail, a relation $R$ on $A$ is reflexive if we have an inclusion
of relations in $\mathbf{Rel}\webleft (A,A\webright )$, i.e. if, for each $a\in A$, we have $a\sim _{R}a$.
Let $A$ be a set.
The set of reflexive relations on $A$ is the subset $\smash {\mathrm{Rel}^{\mathrm{refl}}\webleft (A,A\webright )}$ of $\mathrm{Rel}\webleft (A,A\webright )$ spanned by the reflexive relations.
The poset of relations on $A$ is is the subposet $\smash {\mathbf{Rel}^{\mathsf{refl}}\webleft (A,A\webright )}$ of $\mathbf{Rel}\webleft (A,A\webright )$ spanned by the reflexive relations.
Let $R$ and $S$ be relations on $A$.
