A totally ordered set $X$ is said to be well ordered if every nonempty subset $S$ of $X$ has a smallest element.
Here I am asking about an analogous definition: If every nonempty subset $S$ of $X$ has a largest element, what do we call $X$?
I have heard a lot about well-orderness of sets and am wondering if the "symmetrical" property that every nonempty subset has a largest element is equally important and deserves a familiar name.
In my experience, this is most commonly called "co-well-ordered". Other terms you might encounter are "reverse well-ordered", "anti-well-ordered", or "converse well-ordered".