User blog comment:Edwin Shade/Can Chess Ordinals Produce Functions With Uncountable Growth Rates ?/@comment-1605058-20171222153040/@comment-5529393-20171222230136

A partial ordering is binary relation on a set S (let us denote it with <=) such that:

1. for all a in S, a <= a 2. for all a and b in S, a <= b and b <= a implies a = b 3. for all a, b, and c in S, a <= b and b <= c implies a <= c

Most of the time you see <= used, it refers to a partial ordering.

A well-ordering <= on S is a partial ordering such that any nonempty subset T of S has a minimum element, i.e. there exists a t in T such that, for all s in S, t <= s. Note that this implies that the set is a linear order (for all a,b in S, either a <= b or b <= a); if we have a partial order <= and two elements a,b with neither a <= b nor b <= a holding, then the set {a,b} will not have a minimum element. So all well-orderings are linear orderings, but not vice versa. The integers are a linear order that is not a well-order, since we can take the entire set of integers, and they do not have a minimum element.