Greatest and Least Element

GR#DEF. Greatest and Least Element.

Let $X$ be a preordered set and $a\in X$.

• $a$ is a greatest element of $X$ if for all $x\in X$,

  \[x \leq a.\]

• $a$ is a least element of $X$ if for all $x\in X$,

  \[a \leq x.\]

GR#PROP-MA.

1. Any greatest element is maximal.
2. Any least element is minimal.