I came across the term 'smallest equivalence relation' in the course of a proof I was working on. I have never thought about ordering relations. I googled the term and checked stackexchange and couldn't find a clear definition.
Is someone able to provide me a definition of what a smallest equivalence relation is and an example of one equivalence relation being smaller than another?
Best Answer
I’m going to guess that the actual context was more like the smallest equivalence relation on $A$ satisfying such-and-so conditions. If that’s the case, what is intended is the intersection of all equivalence relations on $A$ satisfying the given conditions: the intersection of equivalence relations on $A$ is an equivalence relation on $A$, and this intersection is the smallest subset of $A\times A$ (smallest in the sense of $\subseteq$) that is an equivalence relation on $A$ and satisfies the given conditions.