---
tag: 99DVVV7BH2
kind: concept
name: Atomistic lattice
msc: [06B05]
specializes: [5SE8EMSDNY]
sources:
  - url: https://en.wikipedia.org/wiki/Atom_(order_theory)
---

## Summary

A lattice with a least element is atomistic if every element is the join of the [[atoms|TTVCTW6TNC]] below it.
So the whole lattice is generated from its atoms by joins, as every subset of a set is the union of its one-element subsets.
Every atomistic lattice is [[atomic|5SE8EMSDNY]], but not conversely.
Geometric lattices, such as the lattice of subspaces of a vector space, are atomistic.

## Explanations

### L1 human

In an atomistic lattice the atoms are building blocks, and every element is put together from the atoms it contains.
In the subsets of a set, every set is the union of its points.
In the subspaces of a finite-dimensional vector space, the atoms are the lines through the origin, and every subspace is spanned by the lines it contains.
A chain $0 < 1 < 2$ is atomic but not atomistic: its only atom is $1$, and $2$ is not a join of atoms.