# Definition:Bottom (Lattice Theory)

## Definition

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice.

### Definition 1

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

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

### Definition 2

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

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

## Equivalence of Definitions

These definitions are shown to be equivalent in Equivalence of Definitions of Bottom.