... Given a totally ordered set (S, ≤) and totally ordered substructures of (S, ≤), we denote the ordering as x ≤ S y (or equivalently, y ≥ S x) when comparing an element x in one substructure with an element y in another substructure with respect to (S, ≤). For example, if (S, ≤) = N and there exist two substructures of N, namely ({1, 2}, ≤) and ({3, 4}, ≤), where ≤ is given by {(1, 1), (1,2), (2, 2)} for the first substructure and { (3,3), (3,4), (4,4)} for the second, then it follows that 3 ≤ N 4. By adopting this notation, we define a totally ordered family of sets (F, ⪯) over (S, ≤). The element-wise ordering axiom can be interpreted as follows: For any two totally ordered sets A and B in a family of sets (F, ⪯), if every element x in A is less than or equal to every element y in B regarding a totally ordered set (S, ≤), then the set A is considered less than or equal to the set B (denoted A ⪯ B). ...