Definition of dicrete ordering

Question

What means by "discrete ordering"?

Answer 1

An ordering is discrete if it is discrete when given the order topology (with basis of open intervals).

It means that every element except possibly the last one (if it exists) has an immediate successor, and every element except possibly the first one has an immediate predecessor.

In general, for any topological property $P$, an ordering has $P$ if it has $P$ when given the order topology (though sometimes other topologies are considered, depending upon the context, e.g. the initial topology or the terminal topology, but most likely not in this case).