---
tag: 5GW2T2AM1A
kind: concept
name: Lattice ideal
msc: [06B10]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_(order_theory)
---

## Summary

An ideal of a [[lattice|WH4SFAACAS]] is a nonempty subset that is closed downward and closed under joins.
The typical example is the principal ideal $\downarrow a = \set{x : x \le a}$ of all elements below a fixed element $a$.
Lattice ideals are the order-theoretic analogue of [[ideals|096Z9MP2ME]] of a ring, and the dual notion is the [[filter|Z15FXM9WM1]].
Prime ideals of lattices are the key to representation theorems for distributive lattices and Boolean algebras.

## Explanations

### L1 human

An ideal of a lattice is a collection of "small" elements.
Anything below a small element is small, and the join of two small elements is still small.
In the subsets of the natural numbers, the finite subsets form an ideal: a subset of a finite set is finite, and the union of two finite sets is finite.
In the positive integers ordered by divisibility, the divisors of $12$ form an ideal.