Let be a poset.
- An element is a maximal element of if there isn’t any such that
- An element is a minimal element of if there isn’t any such that
- doesn’t have any minimal or maximal elements.
- has as its minimal element (if )
- has as its minimal element
- has as its minimal and as its maximal
- has as its minimals, and as its maximals
We have can only have a unique maximal element in totally ordered sets.
There can also be absolute minimal and maximal elements in posets.