What is the supremum and infimum of equivalence relations?

Question

Suppose, I am working on $Eq(X)$ = the set of all equivalence relations on $X$. Let $\theta_1, \theta_2$ denote arbitrary equivalence relations.

Then how does $\theta_1 \lor \theta_2$ look like? And what about $\theta_1 \land \theta_2$?

I know lattices and that these usually correspond to supremum and infimum. But I cannot wrap my head around what does this mean for equivalence relations.

What does $\theta_1 \lor \theta_2$ mean for arbitrary $a, b \in \theta_1 \land \theta_2$?

Thank you.

Answer 1

Recall that an equivalence relation over $X$ is a subset of $X\times X$. In this case the order is containment.

The meet $\land$ is going to be the intersection as it usually works out. Note that the intersection of the relations is symmetric, transitive and reflexive.

The join $\lor$ is going to be be harder to describe, but it's essentially going to be the transitive closure of the union, since transitivity is the thing that can break when you take the union.

Answer 2

If we take two arbitrary elements $a, b$, how does $a (\theta_1 \lor \theta_2) b$ look like?

If this holds, there must exist an element $c$ such that

$a \: \theta_1 \: c$ or $c \: \theta_2 \: b$. (1)

Notice that this is not equivalent to

$a \: \theta_1 \: b$ or $a \: \theta_2 \: b$, (2)

since the join is not the same thing as union. If the union was a transitive relation, the (1) and (2) would be the same thing.