Definition:Bottom of Lattice

Definition

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

Also known as

The bottom of a lattice is also known as a null.

Some sources denote this bottom or null by $0$.

Also see

• Equivalence of Definitions of Bottom of Lattice
• Bottom of Lattice is Unique

• Definition:Top of Lattice

• Results about the bottom of a lattice 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