My textbook gives an example for finding maximal and minimal elements on a set. The set is $(\{2,4,5,10,12,20,25\},|)$. It says to draw a Hasse diagram to find the maximal and minimal elements of the set, saying that the elements on the "top" of the diagram are the maxima, and the ones on the bottom are minima. From the diagram, the author tries to intimate that there can be more than one maximum and minimum. My question is, if we didn't draw this graph, how would you know there are more than one? The definition they provide for maximum and minimums seem to suggest that there can only be one of each.
Here is the author's discussion on this topic, "That is, $a$ is maximal in the poset $(S,\preceq)$ if there is no $b∈S$ such that $a≺b$. Similarly, an element of a poset is called minimal if it is not greater than any element of the poset. That is, $a$ is minimal if there is no element $b∈S$ such that $b≺a$.
Author may be trying to emphasize the following,
Consider the collection of sets $\{\{1, 2, 3\}, \{4\}, \{5, 6\}, \{1, 2, 3, 5,6\}\}$. The minimal members of this collection are $\{1, 2, 3\}, \{4\}, \{5, 6\}$, i.e., These do not contain proper subsets which are members of this collection. The maximal members of this collection are {4}, {1, 2, 3, 5, 6}, i.e., these are not proper subsets of other sets which are members of this collection.
Now build poset for the set in this example and and the point made by author should be clear.
This example is from 'Submodular function and Electrical network', H. Narayanan, p22.