Atom

Remarks

Equivalently, $a$ is an atom when $a$ covers $0$, or when $a$ is a minimal element of $P \setminus \set{0}$. Example: in $\N_{>0}$ ordered by divisibility, with least element $1$, every prime $p$ is an atom. Indeed $1 \ne p$, and the only divisors of $p$ are $1$ and $p$, so nothing lies strictly between them. Non-example: $4$ is not an atom of this poset, since $1 < 2 < 4$ in the divisibility order, so clause 2 fails. The interval $[0, 1]$ with its usual order has no atoms, because $0 < a / 2 < a$ for each $a > 0$. The dual notion, an element just below a greatest element, is a coatom.