Definition:Atom of Lattice

Definition

Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.

An atom of $\struct {S, \vee, \wedge, \preceq}$ is an element $A \in S$ such that:

$\forall B \in S: B \preceq A, B \ne A \implies B = \bot$

$A \ne \bot$

where $\bot$ denotes the bottom of $\struct {S, \vee, \wedge, \preceq}$.

Some sources omit the stipulation that an atom of a lattice is not the bottom of that lattice.

By definition of bottom, there exist no $B$ such that $B \prec \bot$.

Hence $B = \bot$ is vacuously true.

Hence:

$\forall B \in S: B \prec \bot \implies B = \bot$

That is, without explicitly stating that an atom is not the bottom, $\bot$ would be classified as an atom vacuously.

Let $\powerset S$ denote the power set of a set $S$.

Let $\struct {\powerset S, \subseteq}$ be the relational structure defined on $\powerset S$ by the subset relation $\subseteq$.

From Power Set is Lattice, $\struct {\powerset S, \subseteq}$ is a lattice.

The atoms of $\struct {\powerset S, \subseteq}$ are the singleton subsets of $S$.