---
tag: ZHRQ0AT8PM
kind: concept
name: Join-irreducible element
msc: [06B05]
sources:
  - url: https://en.wikipedia.org/wiki/Join_and_meet
---

## Summary

An element of a [[lattice|WH4SFAACAS]] is join-irreducible if it is not the bottom element and not the [[join|J2T75ATRV5]] of two smaller elements.
In a finite lattice, every element is a join of join-irreducible elements, so these elements generate the lattice by joins.
For finite [[distributive lattices|0HM0HQCK6N]], the poset of join-irreducible elements determines the whole lattice, which is the content of Birkhoff's representation theorem.
Meet-irreducible elements are defined dually.

## Explanations

### L1 human

A join-irreducible element is one that cannot be split into two smaller pieces by a join.
In the subsets of a set, the join-irreducible elements are the one-element sets: a set with two or more points is the union of two smaller sets.
In the positive integers ordered by divisibility, where join is the least common multiple, they are the prime powers $p^k$ with $k \ge 1$.
For instance $12 = \lcm(4, 3)$ is not join-irreducible, but $8$ is, since every proper divisor of $8$ divides $4$.