Sublattice of an Atomic Lattice

Question

Is the sublattice of an atomic lattice atomic? If not, then what additional condition is necessary for a sublattice to be atomic?

Answer 1

Theorem. The following are equivalent for a lattice $L$.

1. Every sublattice of $L$ is atomic.
2. $L$ satisfies the descending chain condition (DCC).

[A lattice satisfies the DCC iff it has no infinite strictly descending chain iff any nonempty subset of $L$ has a minimal element. Every finite lattice satisfies the DCC, since a finite lattice cannot contain an infinite strictly decreasing chain.]

Reasoning.
Assume that $K$ is a lattice that satisfies the DCC. For any $x\in K-\{0\}$, the nonempty half-open interval $(0,x]$ must have a minimal element $y$. Necessarily $y$ is an atom of $K$ below $x$. Thus, if $K$ satisfies the DCC, then $K$ is atomic.

Now assume that Item $2$ of the theorem is true for some lattice $L$. Since any sublattice of a DCC lattice also satisfies the DCC, any sublattice of $L$ is atomic. Hence Item $1$ of the theorem is true.

Next, assume that Item $2$ fails, then $L$ contains an infinite subset $C = \{x_0, x_1, \ldots\}\subseteq L$ with $x_0>x_1>\cdots$.
$C$ is a sublattice of $L$ that has no atoms, so it is a nonatomic sublattice of $L$. Hence Item $1$ fails. \\\