What is the generalization of well-orders to partial orders?

Question

Every well-ordered set is linearly ordered. However, is there a notion of well-orders that generalizes to partial orders? Maybe, the generalized definition will be that every non-empty subset has a minimal element, not necessarily a least element. I would be very interested to hear about such a definition, and more importantly, I would like to know about research on this topic about these "generalized" well-orders.

Answer 1

There are at least three notions that are potentially of interest. The simplest generalization is well-founded partial order: this is just a partial order with no descending chain. Equivalently, a partial order in which every nonempty set has at least one minimal element. Almost always this is the "right" generalization to work with.

However, there are two other definitions that are worth mentioning. First, there are well-quasi-orders. A WQO is a partial order in which there are no infinite descending chains or antichains; for example, the partial order of strings of natural numbers with respect to extension is well-founded but is not a WQO. WQOs in turn lead to a yet-more-technical notion, that of a better quasi-order. Roughly speaking, BQOs arise as a way to strengthen an induction hypothesis: often when we want to show that a certain partial $P$ order built from a "starting WQO $X$ is a WQO, it's actually easier to show that $X$ is a BQO and therefore so is $P$. The definition of BQO is rather complicated; a good introduction to WQO and BQO theory is given in Rosenstein's sadly-very-out-of-print book Linear Orderings.