TITLE: Definition of smallest equivalence relation
QUESTION [12 upvotes]: 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?

REPLY [5 votes]: Equivalence relations are (partially) ordered by implication; $\Theta \leq \Phi$ if and only if
$$ x \Theta y \implies x \Phi y $$
is an identity.
In fact, this partial ordering is a complete lattice; the meet operation (a.k.a. the greatest lower bound) is given by logical and. That is, the equivalence relation $\Theta = \bigwedge_{i \in I} \Theta_i$ is the one defined by
$$x \Theta y \Longleftrightarrow \bigwedge_{i \in I} x \Theta_i y$$
or in terms of quantifiers rather than infinitary operations,
$$x \Theta y \Longleftrightarrow \forall i \in I: x \Theta_i y $$
If you look at the graph of the relations, this can all be rephrased in terms of subsets and intersections.

A simple source of examples is to use the first isomorphism theorem — there is a bijective correspondence between congruences and quotients.
e.g. in terms of sets, quotients can be viewed as partitions of the set, and the correspondence is that the equivalence classes of an equivalence relation are the parts of the partition.
The smallest equivalence relation on a nonempty set $X$ corresponds to the partition whose classes are all singletons; it is equality. The largest equivalence relation is the one with just a single partition; the corresponding equivalence relation is the one where everything is related. The ordering on equivalence relations corresponds to whether one partition refines another.