Greatest element and least element

In mathematics, especially in order theory, the greatest element of a subset ${\displaystyle S}$ of a partially ordered set (poset) is an element of ${\displaystyle S}$ that is greater than every other element of ${\displaystyle S}$. The term least element is defined dually, that is, it is an element of ${\displaystyle S}$ that is smaller than every other element of ${\displaystyle S.}$

Let ${\displaystyle (P,\leq )}$ be a preordered set and let ${\displaystyle S\subseteq P.}$ An element ${\displaystyle g\in P}$ is said to be a greatest element of ${\displaystyle S}$ if ${\displaystyle g\in S}$ and if it also satisfies:

By switching the side of the relation that ${\displaystyle s}$ is on in the above definition, the definition of a least element of ${\displaystyle S}$ is obtained. Explicitly, an element ${\displaystyle l\in P}$ is said to be a least element of ${\displaystyle S}$ if ${\displaystyle l\in S}$ and if it also satisfies:

If ${\displaystyle (P,\leq )}$ is also a partially ordered set then ${\displaystyle S}$ can have at most one greatest element and it can have at most one least element. Whenever a greatest element of ${\displaystyle S}$ exists and is unique then this element is called the greatest element of ${\displaystyle S}$. The terminology the least element of ${\displaystyle S}$ is defined similarly.

If ${\displaystyle (P,\leq )}$ has a greatest element (resp. a least element) then this element is also called a top (resp. a bottom) of ${\displaystyle (P,\leq ).}$

Greatest elements are closely related to upper bounds.

Let ${\displaystyle (P,\leq )}$ be a preordered set and let ${\displaystyle S\subseteq P.}$ An upper bound of ${\displaystyle S}$ in ${\displaystyle (P,\leq )}$ is an element ${\displaystyle u}$ such that ${\displaystyle u\in P}$ and ${\displaystyle s\leq u}$ for all ${\displaystyle s\in S.}$ Importantly, an upper bound of ${\displaystyle S}$ in ${\displaystyle P}$ is not required to be an element of ${\displaystyle S.}$

If ${\displaystyle g\in P}$ then ${\displaystyle g}$ is a greatest element of ${\displaystyle S}$ if and only if ${\displaystyle g}$ is an upper bound of ${\displaystyle S}$ in ${\displaystyle (P,\leq )}$ and ${\displaystyle g\in S.}$ In particular, any greatest element of ${\displaystyle S}$ is also an upper bound of ${\displaystyle S}$ (in ${\displaystyle P}$) but an upper bound of ${\displaystyle S}$ in ${\displaystyle P}$ is a greatest element of ${\displaystyle S}$ if and only if it belongs to ${\displaystyle S.}$ In the particular case where ${\displaystyle P=S,}$ the definition of "${\displaystyle u}$ is an upper bound of ${\displaystyle S}$ in ${\displaystyle S}$" becomes: ${\displaystyle u}$ is an element such that ${\displaystyle u\in S}$ and ${\displaystyle s\leq u}$ for all ${\displaystyle s\in S,}$ which is completely identical to the definition of a greatest element given before. Thus ${\displaystyle g}$ is a greatest element of ${\displaystyle S}$ if and only if ${\displaystyle g}$ is an upper bound of ${\displaystyle S}$ in ${\displaystyle S}$.