Lattice of congruences of a lattice is algebraic

Question

I'm trying to understand some basics of the lattice theory and I would like to know how one can prove that given any lattice L the (complete) lattice of all congruences on L is an algebraic lattice.

I know the "increasing chain" charecterisation of the join operation in the lattice of congruences on a lattice, but I don't know how to apply it.

I'll be grateful for any help.

Answer 1

You can use these facts:

1. every congruence $\theta$ of $\mathbf{L}$, $$\theta = \bigcup_{\langle a, b \rangle \in \theta} \Theta(a,b) = \bigvee_{\langle a, b \rangle \in \theta} \Theta(a,b),$$ where $\Theta(a,b)$ is the least congruence of $\mathbf{L}$ that contains the pair $\langle a, b \rangle$.

2. a compact congruence of $\mathbf{L}$ is a finitely generated one.

Perhaps you still have to prove this last fact, but it is not difficult; the first one is trivial.