Bottom of Lattice is Unique

Theorem

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

Then $S$ has at most one bottom.

Proof

By definition, a bottom for $S$ is a smallest element.

The result follows from Smallest Element is Unique.

$\blacksquare$

Also see

• Top of Lattice is Unique