Dan Schmidt wrote:
> Following Definition 1.60 in the book, \\(p\\) is the meet of \\(\emptyset\\) if
> 1. for all \\(a \in \emptyset\\), \\(p \leq a\\), which imposes no restrictions at all, since there is no such \\(a\\);
> 2. for all \\(q\\) such that \\(q \leq a\\) for all \\(a \in A\\), it is true that \\(q \leq p\\). Well, the antecedent is true of every \\(q\\), since no restriction is placed on it. So \\(q \leq p\\) for all \\(q\\) in the entire poset, and \\(p\\) must satisfy this if possible. For example, if the poset is a power set \\(PX\\) with the inclusion relation, \\(p\\) is the entire set \\(X\\).
Good! So, for any poset \\(A\\), the meet of \\(\emptyset \subseteq A \\) in , if it exists, is the **top element** of \\(A\\): the element \\(p \in A\\), necessarily unique, such that \\(q \le p\\) for all \\(q \in A\\).
The top element is usually denoted \\(\top\\). We draw it at the top of a Hasse diagram, like this Hasse diagram of the poset of 4-bit strings: