A relation $R$ is an equivalence relation if it is reflexive, symmetric, and transitive.1
- 1Further Terminology: If instead $R$ is just symmetric and transitive, then it is called a partial equivalence relation.
10.5.1 Foundations
Let $A$ be a set.
The kernel of a function $f\colon A\to B$ is the equivalence relation $\mathord {\sim }_{\mathrm{Ker}\webleft (f\webright )}$ on $A$ obtained by declaring $a\sim _{\mathrm{Ker}\webleft (f\webright )}b$ iff $f\webleft (a\webright )=f\webleft (b\webright )$.1
-
1The kernel $\mathrm{Ker}\webleft (f\webright )\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ of $f$ is the underlying functor of the monad induced by the adjunction $\operatorname {\mathrm{Gr}}\webleft (f\webright )\dashv f^{-1}\colon A\mathbin {\rightleftarrows }B$ in $\boldsymbol {\mathsf{Rel}}$ of Chapter 9: Constructions With Relations,
of .
Let $A$ and $B$ be sets.
-
1.
The set of equivalence relations from $A$ to $B$ is the subset $\smash {\mathrm{Rel}^{\mathrm{eq}}\webleft (A,B\webright )}$ of $\mathrm{Rel}\webleft (A,B\webright )$ spanned by the equivalence relations.
-
2.
The poset of relations from $A$ to $B$ is is the subposet $\smash {\mathbf{Rel}^{\operatorname {\mathrm{eq}}}\webleft (A,B\webright )}$ of $\mathbf{Rel}\webleft (A,B\webright )$ spanned by the equivalence relations.
