---
tag: J5HQWT3ZDA
kind: definition
name: Lattice ideal
msc: [06B10]
defines: 5GW2T2AM1A
uses: [WH4SFAACAS, J2T75ATRV5]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_(order_theory)
---

## Definition

Let $(L, \vee, \wedge)$ be a [[lattice|WH4SFAACAS]] with its order $\le$.
An *ideal* of $L$ is a subset $I \subseteq L$ such that:

1. $I$ is nonempty;
2. if $x \in I$, $y \in L$ and $y \le x$, then $y \in I$;
3. if $x, y \in I$, then $x \vee y \in I$.

The ideal $I$ is *proper* if $I \ne L$.

## Remarks

For $a \in L$, the set $\downarrow a = \set{x \in L : x \le a}$ is an ideal, the *principal ideal* generated by $a$.
In a finite lattice every ideal is principal, generated by the join of all its elements.
Example: in the power set of $\N$ ordered by $\subseteq$, the finite subsets form an ideal.
It is nonempty, every subset of a finite set is finite, and the union of two finite sets is finite.
It is not principal, since no finite set contains all finite sets.
Non-example: in the power set of $\set{1, 2}$, the set $\set{\emptyset, \set{1}, \set{2}}$ satisfies clauses 1 and 2 but not clause 3, since $\set{1} \cup \set{2} = \set{1, 2}$ is missing.
The same notion makes sense in any [[partially ordered set|0QV2JS9RJ3]], with clause 3 replaced by: any two elements of $I$ have an upper bound in $I$.
In a lattice the two versions agree.