Equivalence of Definitions of Top of Lattice

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

Then the following definitions of $\top$, top, are equivalent:

Proof
By definition, $\top$ is the greatest element of $S$ iff 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.

Also see

 * Top