How can I show that a lattice $(S, \leq)$ must have a greatest element?

Question

How can I show that a lattice $(S, \leq)$, where $S$ is a finite set, must have a greatest element? What I mean is an element $x$ such that $a \leq x$ for all $a \in S$.

Answer 1

Hint:

Based on the definition of a lattice it can be shown that - if a lattice $S$ has no greatest element - for every $a\in S$ there will exist some $b\in S$ with $a\leq b$ and $a\neq b$.

Now observe that this cannot occur in a finite lattice.

Answer 2

You can build the greatest element by induction.

If the lattice has $1$ element, of course the property is trivial.

Suppose you have a maximum for lattices with $n$ elements. Then take a lattice with $n+1$ elements $\{x_0,...,x_n\}$. By hypothesis of induction $$a = x_0 \vee ... \vee x_{n-1} $$ exists. Then $$a \vee x_n$$ is the greatest element of your lattice.