---
tag: 9H61HXXJ11
kind: concept
name: Join-prime element
msc: [06B05]
specializes: [ZHRQ0AT8PM]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_element
---

## Summary

An element $a$ of a [[lattice|WH4SFAACAS]] is join-prime if it is not the bottom element and has the following property.
Whenever $a$ lies below a join $b \vee c$, it lies below $b$ or below $c$.
The definition copies that of a [[prime element|01VH21EV6F]] of a ring, with "divides" replaced by "lies below" and the product replaced by the [[join|J2T75ATRV5]].
Every join-prime element is [[join-irreducible|ZHRQ0AT8PM]], and in a [[distributive lattice|0HM0HQCK6N]] the two notions coincide.

## Explanations

### L1 human

A join-prime element cannot hide inside a join without hiding inside one of the two parts.
In the subsets of a set, a singleton $\set{x}$ is join-prime: if $x$ is in $B \cup C$, then $x$ is in $B$ or in $C$.
In the lattice $\set{0, a, b, c, 1}$ with three incomparable middle elements, $a$ is join-irreducible but not join-prime.
It lies below $b \vee c = 1$ but not below $b$ or $c$.