Atomic lattice

Remarks

The same definition applies to any partially ordered set with a least element. Example: every finite lattice $L$ is atomic. For $b \ne 0$, the set $\set{x \in L : 0 < x \le b}$ is finite and nonempty, so it has a minimal element, and a minimal element of this set is an atom. Example: $\N_{>0}$ ordered by divisibility is atomic, since every $n > 1$ has a prime divisor. Non-example: the chain $[0, 1]$ with its usual order is a lattice with least element $0$ and no atoms, so the condition fails for $b = 1$.