Lattice congruence

Remarks

The quotient set $L / \theta$ is a lattice with $[x] \vee [y] = [x \vee y]$ and $[x] \wedge [y] = [x \wedge y]$; clauses 1 and 2 say that these operations are well defined. The kernel $\set{(x, y) : f(x) = f(y)}$ of a lattice homomorphism $f$ is a congruence, and every congruence $\theta$ is the kernel of the quotient map $L \to L / \theta$. Each class of a congruence is a convex sublattice of $L$. Example: on the chain $0 < 1 < 2$, the partition $\set{0}, \set{1, 2}$ is a congruence. It is the kernel of the monotone map $0 \mapsto 0$, $1 \mapsto 1$, $2 \mapsto 1$ onto the chain $0 < 1$. This map is a lattice homomorphism because join and meet in a chain are $\max$ and $\min$. Non-example: on the lattice $\set{0, a, b, 1}$ with $0 < a, b < 1$ and $a$, $b$ incomparable, the partition $\set{0, 1}, \set{a}, \set{b}$ is not a congruence. From $0 \mathrel{\theta} 1$ and $a \mathrel{\theta} a$, clause 1 would give $(0 \vee a) \mathrel{\theta} (1 \vee a)$, that is $a \mathrel{\theta} 1$, which is false.