---
tag: 82D3DQD3X8
kind: definition
name: Sublattice
msc: [06B05]
defines: PV90EW00XK
uses: [WH4SFAACAS, J2T75ATRV5]
sources:
  - url: https://en.wikipedia.org/wiki/Lattice_(order)
---

## Definition

Let $(L, \vee, \wedge)$ be a [[lattice|WH4SFAACAS]].
A *sublattice* of $L$ is a nonempty subset $M \subseteq L$ such that $x \vee y \in M$ and $x \wedge y \in M$ for all $x, y \in M$.

## Remarks

A sublattice $M$ is a lattice with the restricted operations, and its order is the order of $L$ restricted to $M$.
Some authors allow the empty subset as a sublattice.
Example: in the power set of $\set{1, 2, 3}$ ordered by $\subseteq$, the set $M = \set{\emptyset, \set{1}, \set{2}, \set{1, 2}}$ is a sublattice.
The union and the intersection of two members of $M$ are subsets of $\set{1, 2}$, and all subsets of $\set{1, 2}$ are in $M$.
Non-example: $N = \set{\emptyset, \set{1}, \set{2}, \set{1, 2, 3}}$ is a lattice in the induced order, where the join of $\set{1}$ and $\set{2}$ is $\set{1, 2, 3}$.
It is not a sublattice, because the join of $\set{1}$ and $\set{2}$ in the power set is $\set{1, 2} \notin N$.
Every chain in a lattice is a sublattice, since $x \vee y$ and $x \wedge y$ are $x$ or $y$ when $x$ and $y$ are comparable.