Equivalence of Definitions of Top of Lattice

Theorem

The following definitions of the concept of Top of Lattice are equivalent:

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

Definition 1

Let $S$ admit a greatest element $\top$.

Then $\top$ is called the top of $S$.

Definition 2

Let $\wedge$ have an identity element $\top$.

Then $\top$ is called the top of $S$.

Proof

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

$a \preceq \top$

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

$a \wedge \top = a$

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

The result follows.

$\blacksquare$