Maximum and Minimum

Definition 1. Let $X$ be a partially ordered set.

If $X$ has a greatest element, then it is unique and is called the maximum of $X$, denoted by
\[\max X.\]
If $X$ has a least element, then it is unique and is called the minimum of $X$, denoted by
\[\min X.\]

Proof (uniqueness)

If both $a$ and $a’$ are greatest elements of $X$, $a’ \leq a$ and $a \leq a’$, so $a=a’$ by antisymmetry. Similarly for the minimum.

References

[1]: Wikipedia contributors. Maximum and minimum. Wikipedia, The Free Encyclopedia.
[2]: Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. p. 16.