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 $\struct {S, \vee, \wedge, \preceq}$.
Also defined as
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:
- $\forall B \in S: B \prec \bot \implies B = \bot$
is true.
That is, without explicitly stating that an atom is not the bottom, $\bot$ would be classified as an atom vacuously.
Examples
Singleton in Lattice of Sets
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 $S$.
Also see
- Results about atoms of lattices can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): atom
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): atom