Preorder

Overview

A binary relation $R$ on set $A$ is a non-strict preorder on $A$ iff it is reflexive on $A$ and transitive. If $R$ is instead irreflexive on $A$, it is a strict preorder.

Boundedness

Let $P$ be a preordered set with order relation $\le$.

Given some subset $S$ of $P$, an upper bound of $S$ is some $U\in P$ such that $U$ is greater than or equal to every element of $S$. We say $S$ is bounded above by $U$. A lower bound of $S$ is some $L\in P$ such that $L$ is less than or equal to every element of $S$. We say $S$ is bounded below by $L$.

A preordered set with no upper bound is said to be unbounded above. A preordered set with no lower bound is said to be unbounded below.

Maximals and Minimals

Let $P$ be a preordered set with order relation $\le$.

An element $M\in P$ is maximal if there is no other element of $P$ greater than $M$. An element $m\in P$ is minimal if there is no other element of $P$ less than $m$.

Maximums and Minimums

Let $P$ be a preordered set with order relation $\le$.

An element $M\in P$ is maximum, denoted $M=\text{max}P$, if $M$ is greater than or equal to every element of $P$. An element $m\in P$ is minimum, denoted $m=\text{min}P$, if $m$ is less than or equal to every element of $P$.

Extremums

A real-valued function $f$ is said to have an absolute maximum on a set $S$ if there is at least one point $c\in S$ such that $f(x)\le f(x)$ for all $x\in S$. $f$ is said to have an absolute minimum on $S$ if there is at least one point $d\in S$ such that $f(d)\le f(x)$ for all $x\in S$.

$f$ is said to have a relative maximum at a point $c\in S$ if there is an open interval $I$ containing $c$ such that $f(x)\le f(c)$ for all $x\in I\cap S$. Likewise, $f$ is said to have a relative minimum at a point $d\in S$ if there is an open interval $I$ containing $c$ such that $f(d)\le f(x)$ for all $x\in I\cap S$.

An extremum (or extreme value) of $f$ is either a relative maximum or a relative minimum of $f$.

Supremums and Infimums

Let $P$ be a preordered set with order relation $\le$.

Given some subset $S$ of $P$, a member $M\in P$ is a supremum (or least upper bound) of $S$ if $M$ is an upper bound for $S$ and no member less than $M$ is an upper bound for $S$. This is denoted as $M=\text{lub}S$ or $M=\text{sup}S$.

A member $m\in P$ is an infimum (or greatest lower bound) of $S$ if $m$ is a lower bound for $S$ and no member greater than $m$ is a lower bound for $S$. This is denoted as $m=\text{glb}S$ or $m=\text{inf}S$.