atom
Let $P$ be a poset, partially ordered by $\le $. An element $a\in P$ is called an atom if it covers some minimal element of $P$. As a result, an atom is never minimal^{}. A poset $P$ is called atomic if for every element $p\in P$ that is not minimal has an atom $a$ such that $a\le p$.
Examples.

1.
Let $A$ be a set and $P={2}^{A}$ its power set^{}. $P$ is a poset ordered by $\subseteq $ with a unique minimal element $\mathrm{\varnothing}$. Thus, all singleton subsets $\{a\}$ of $A$ are atoms in $P$.

2.
${\mathbb{Z}}^{+}$ is partially ordered if we define $a\le b$ to mean that $a\mid b$. Then $1$ is a minimal element and any prime number $p$ is an atom.
Remark. Given a lattice^{} $L$ with underlying poset $P$, an element $a\in L$ is called an atom (of $L$) if it is an atom in $P$. A lattice is a called an atomic lattice if its underlying poset is atomic. An atomistic lattice is an atomic lattice such that each element that is not minimal is a join of atoms. If $a$ is an atom in a semimodular lattice $L$, and if $a$ is not under $x$, then $a\vee x$ is an atom in any interval lattice $I$ where $x=\bigwedge I$.
Examples.

1.
$P={2}^{A}$, with the usual intersection^{} and union as the lattice operations^{} meet and join, is atomistic: every subset $B$ of $A$ is the union of all the singleton subsets of $B$.

2.
${\mathbb{Z}}^{+}$, partially ordered as above, with lattice binary operations^{} defined by $a\wedge b=\mathrm{gcd}(a,b)$, and $a\vee b=\mathrm{lcm}(a,b)$, is a lattice that is atomic, as we have seen earlier. But it is not atomistic: $4$ is not a join of $2$’s; $36$ is not a join of $2$ and $3$ are just two counterexamples.
Title  atom 

Canonical name  Atom 
Date of creation  20130322 15:20:09 
Last modified on  20130322 15:20:09 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  13 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06A06 
Classification  msc 06B99 
Defines  atomic poset 
Defines  atomic lattice 
Defines  atomistic lattice 
Defines  atomistic 