Why are complete lattices non-empty?

Question

I am confused as to whether complete lattices are empty while reading the following article:

Complete Lattice

A complete lattice is a partially ordered set $(L,\leq)$ in which every subset of L has a greatest lower bound (glb) and a least upper bound (lub) in L. The lub of the empty set is denoted by $\bot$ and glb of the empty set is $\top$. It follows that a complete lattice is never empty.

The author just stated that "The lub of the empty set is ... $\bot$ and glb of the empty set is $\top$.", and the only subset of empty set is itself, so every subset of it does have a lub ($\bot$) and glb ($\top$), and it should be complete according to this definition. I may be missing something obvious, but

Why is it stated that empty set is not complete here and elsewhere?

-- Edit --

From the answers so far, my understanding is that the statement in question was really stating that the underlying poset of a complete lattice is non-empty.

A further question is, is this non-empty assertion unnecessary?

Since a poset is defined on a set with an ordering, the poset is never empty by definition (e.g. even empty the set has a non-empty powerset). Why is it necessary to assert whether a complete lattice is empty or not (if all posets are non-empty in this sense).

Answer 1

The last two sentences of the quoted passage are, in my opinion, out of order. One should first notice that, in a complete lattice $L$, the empty subset must have a supremum and an infimum; these are elements of $L$, so $L$ isn't empty. Then, introduce the notations $\bot$ and $\top$ for the supremum and the infimum of $\varnothing$ (i.e., the bottom and top elements of $L$).

Note that $\bot$ and $\top$ might be equal (if $L$ has only one element), but they (or it) must be present in $L$, so $L$ can't be empty.

Answer 2

Here's what the author meant:

Complete Lattice

A complete lattice is a partially ordered set $(L,\leq)$ in which every subset of $S$ of $L$ has a greatest lower bound (glb) and a least upper bound (lub) in $L$. When $S=\varnothing$ the lub of $S$ is denoted by $\bot$ and glb is denoted by $\top$. It follows that a complete lattice is never empty.

Answer 3

Since the empty subset is always a subset, even of the empty set itself, it then has greatest lower and least upper bounds. Therefore, the lattice set $L$ contains bounds. It contains something. That means it's not an empty set.