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 $a\wedge b=\gcd (a,b)$, 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.