Relations / Poset Extrema
Least You Need to Know: Minimal, Maximal, Least, and Greatest
In a poset, **least** and **greatest** are stronger than **minimal** and **maximal**. Least means below everything; minimal only means nothing is strictly below it.
The least you need to know
- A least element is unique if it exists.
- A poset can have several minimal elements.
- Greatest implies maximal, but not every maximal element is greatest.
- Use the order relation carefully; compare elements, not their sizes in everyday language.
Key notation
x ≤ y
x is below y in the poset
least element
below every element
minimal element
no different element lies below it
Tiny worked example
- In the divisibility poset on `{2,3,6}`, both 2 and 3 are minimal.
- Neither is least because 2 does not divide 3 and 3 does not divide 2.
- The element 6 is maximal, and here it is also greatest.
Common mistakes
- Students often treat minimal as the same as smallest in everyday size.
- Students often forget incomparable elements can create multiple minimal elements.
- Students often assume every finite poset has a least element.
How to recognize this kind of problem
- Ask whether one element is below all others or just has nothing below it.
- If two elements are incomparable, neither can be least unless one is actually below the other.
- Hasse diagrams help visualize extrema.