A poset need not have a first or last element.
- An element $a_1$ is first (the unique minimum) in a poset $P$ if $a_1 \le a\;\;\forall a \in P$.
- An element $a_n$ is last (the unique maximum) in a poset P if $a \le a_n \;\;\forall a \in P$.
(1) There may not be a "minimum" (first) nor "maximum" (last) element in a poset.
(2) You might have a unique minimum (first) element, but no maximum element (last) in a poset.
(3) Likewise, a poset may not have a minimum "first" element, but have a maximum ("last") element.
(4) And if there is a unique minimum and a unique maximum element in the poset, then those are the first and last elements in the poset, respectively.
Exercise: try to find an "example" poset that models/represents each of these four possibilities) (one per possibility)!