well-order

A well-order is a homogeneous relation that is reflexive, transitive, antisymmetric, connected, and has a least element in every subset.

Formally,
a binary relation $\le$ over a set $S$ is a well-order if for all $a,b,c$ in $S$,

• $a\le a$ (reflexive)
• If $a\le b$ and $b\le c$ then $a\le c$ (transitive)
• If $a\le b$ then $b\nleqq a$ (antisymmetric)
• $a\le b$ or $b\le a$ (connected)
• $\mathrm{\exists }!l_W\in W$ such that $l_W\le w$ for all $w$ in every subset $W\subseteq X$ (least element)

A well-order is a total order such that every subset has a least element.