Artur - you're not making a mistake; this is an important observation.

The meet of empty set, if it exists, must be the biggest thing that's smaller than everything in the empty set. But "smaller than everything in the empty set" is a vacuous condition, so the meet of the empty set, if it exists, must be the biggest thing. That is, it must be an element that's greater than or equal to all others. In a poset, this element is unique (if it exists), and we call it the **top** element or \\(\top\\). In the logic of subsets this element is also called **true**.

All this has an upside-down version for joins: the join of the empty set, if it exists, must be an element than's less than or equal to all others; in a poset it is unique (if it exists) and is called the **bottom** element or \\(\bot\\) or **false**.

The meet of empty set, if it exists, must be the biggest thing that's smaller than everything in the empty set. But "smaller than everything in the empty set" is a vacuous condition, so the meet of the empty set, if it exists, must be the biggest thing. That is, it must be an element that's greater than or equal to all others. In a poset, this element is unique (if it exists), and we call it the **top** element or \\(\top\\). In the logic of subsets this element is also called **true**.

All this has an upside-down version for joins: the join of the empty set, if it exists, must be an element than's less than or equal to all others; in a poset it is unique (if it exists) and is called the **bottom** element or \\(\bot\\) or **false**.