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.
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
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 it
- 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.
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