Integers and fractions have the same cardinality (countably infinite) but are different
in that between any two distinct fractions is another fraction (actually an infinite
number of them) but this property does not hold for integers. I am looking for a
mathematical term that captures this property of fractions that integers do not have.
We say that the order relation "<" is dense
on the rationals, but not on the integers.
So, I guess you could say that the rationals are dense as a partially ordered set under the ordinary ordering "<", whereas the integers are not.
(Some readers may be more familiar with the idea that a metric space can be a "dense" subset of another metric space. For example, the rationals are a "dense" subset of the reals, whereas the integers are not. Similar concept. Slightly different use of the same word.)
Note that this property also holds for the set of strings (lexicographically sorted);
that is there are an infinite number of strings between the strings 'abc' and 'defg'.
In other words, the set of all lexicographically sorted character-strings is dense as a partially ordered set.