---
tag: B0QT3SM5DT
kind: concept
name: Bounded poset
msc: [06A06]
specializes: [0QV2JS9RJ3]
sources:
  - url: https://en.wikipedia.org/wiki/Greatest_element_and_least_element
---

## Summary

A bounded poset is a [[partially ordered set|011C29Q1S1]] that has both a [[greatest element|0Z7HDDXX3X]] and a [[least element|0T2NTJCV83]].
These are called top and bottom and written $\top$ and $\bot$, or $1$ and $0$.
Bounded posets are the setting for complements in [[lattices|WH4SFAACAS]].

## Explanations

### L1 human

A bounded poset has a top element above everything and a bottom element below everything.
The subsets of a set, ordered by inclusion, form a bounded poset: the empty set is the bottom and the whole set is the top.
The closed interval $[0, 1]$ with its usual order is bounded, with bottom $0$ and top $1$.
The natural numbers are not bounded: they have a bottom, $0$, but no top.