If $a$ is an element in a preorder then you can eventually go 'a step back' (and repeat this) in the sense of finding an element with $b\leq a$ and not $a\leq b$.
Is there a special name for (pre)orders in which for every element such a route back can only contain a finite number of steps?
This question occurred when I was looking at factorization in irreducibles in an integral domain.