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 $\mathord {\sim }_{\mathrm{Ker}(f)}$ on $A$ obtained by declaring $a\sim _{\mathrm{Ker}(f)}b$ iff $f(a)=f(b)$.1
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1The kernel $\mathrm{Ker}(f)\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ of $f$ is the underlying functor of the monad induced by the adjunction $\operatorname {\mathrm{Gr}}(f)\dashv f^{-1}\colon A\mathbin {\rightleftarrows }B$ in $\boldsymbol {\mathsf{Rel}}$ of Chapter 9: Constructions With Relations,
of .
Let $A$ and $B$ be sets.
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1.
The set of equivalence relations from $A$ to $B$ is the subset $\smash {\mathrm{Rel}^{\mathrm{eq}}(A,B)}$ of $\mathrm{Rel}(A,B)$ spanned by the equivalence relations.
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2.
The poset of relations from $A$ to $B$ is is the subposet $\smash {\mathbf{Rel}^{\operatorname {\mathrm{eq}}}(A,B)}$ of $\mathbf{Rel}(A,B)$ spanned by the equivalence relations.
