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: