Definition 1. Let $X$ be a preordered set and $a \in X$.
-
$a$ is a greatest element of $X$ if
\[\forall x \in X : x \leq a.\] -
$a$ is a least element of $X$ if
\[\forall x \in X : a \leq x.\]
- Any greatest element is maximal.
- Any least element is minimal.
References
- [1]
- Wikipedia contributors. Greatest element and least element. Wikipedia, The Free Encyclopedia.
- [2]
- Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. p. 16.