---
tag: V465WCM25T
kind: definition
name: Lattice congruence
msc: [06B10, 08A30]
defines: T782PJYVEZ
uses: [WH4SFAACAS, 0G8VEZ3WTK, BNVRR925X8]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence_relation
---

## Definition

Let $(L, \vee, \wedge)$ be a [[lattice|WH4SFAACAS]].
A *congruence* on $L$ is an [[equivalence relation|0G8VEZ3WTK]] $\theta$ on $L$ such that, for all $x, y, x', y' \in L$ with $x \mathrel{\theta} y$ and $x' \mathrel{\theta} y'$:

1. $(x \vee x') \mathrel{\theta} (y \vee y')$;
2. $(x \wedge x') \mathrel{\theta} (y \wedge y')$.

## 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|BNVRR925X8]] $f$ 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.