Equivalence of Definitions of Bottom of Lattice

Theorem

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

Let $\bot$ be a bottom of $\struct {S, \vee, \wedge, \preceq}$.

The following definitions of the concept of Bottom in the context of Lattice Theory are equivalent:

Let $S$ admit a smallest element $\bot$.

Then $\bot$ is called the bottom of $S$.

Let $\vee$ have an identity element $\bot$.

Then $\bot$ is called the bottom of $S$.

By definition, $\bot$ is the smallest element of $S$ if and only if for all $a \in S$:

$\bot \preceq a$

By Ordering in terms of Join, this is equivalent to:

$a \vee \bot = a$

If this equality holds for all $a \in S$, then by definition $\bot$ is an identity for $\vee$.

The result follows.

$\blacksquare$