---
tag: 72V1Z82M5F
kind: concept
name: Bounded lattice
msc: [06B05]
specializes: [WH4SFAACAS]
sources:
  - url: https://en.wikipedia.org/wiki/Bounded_lattice
---

## Summary

A bounded lattice is a [[lattice|WH4SFAACAS]] with a [[greatest element|0XR2QK49V6]] $1$ and a [[least element|0AANKKNHS4]] $0$.
These elements are the identities for the meet and the join: $x \wedge 1 = x$ and $x \vee 0 = x$.
Every finite nonempty lattice is bounded, and bounded lattices are the setting for complements, atoms and Boolean algebras.

## Explanations

### L1 human

A bounded lattice has a bottom element below everything and a top element above everything.
In the lattice of subsets of a set $X$, the bottom is the empty set and the top is $X$ itself.
The interval $[0, 1]$ with its usual order is a bounded lattice, with bottom $0$ and top $1$.
The integers with their usual order form a lattice that is not bounded: there is no largest and no smallest integer.