# Equivalence of Definitions of Top

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## Theorem

The following definitions of the concept of **$\top$ (top)** in the context of **Lattice Theory** are equivalent:

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

$\blacksquare$