---
tag: PV90EW00XK
kind: concept
name: Sublattice
msc: [06B05]
sources:
  - url: https://en.wikipedia.org/wiki/Lattice_(order)
---

## Summary

A sublattice of a [[lattice|WH4SFAACAS]] $L$ is a nonempty subset of $L$ that is closed under the [[join and meet|J2T75ATRV5]] of $L$.
It is a lattice in its own right, with the operations of $L$.
Sublattices play the role that subgroups play for groups, and forbidden sublattices characterise classes of lattices, such as the distributive ones.

## Explanations

### L1 human

Take a lattice and keep only some of its elements.
The kept part is a sublattice when the join and the meet of any two kept elements, computed in the big lattice, are again kept.
For example, in the subsets of $\set{1, 2, 3}$, the four sets $\emptyset$, $\set{1}$, $\set{2}$, $\set{1, 2}$ form a sublattice.
A subset can be a lattice in its own order without being a sublattice, because its joins may differ from those of the big lattice.