One example would be if you define one set A to be "less than" another B if A is a subset of B. Then ∅ < {0} and {0} < {0, 1} but {0} ≹ {1}.

Such a thing is called a partial ordering and a set of values with a partial ordering is called a partially ordered set or poset (pronounced Poe-set) for short.

https://en.wikipedia.org/wiki/Partially_ordered_set