ordering relation

A partial ordering on a set is a relation < that is transitive and reflexive and antisymmetric. That is, (i) x < y & y < z → x < z; (ii) x < x; (iii) x < y & y < x → x = y. If we add (iv) that at least one of x < y, x = y, and y < x holds (the relation is connected, or, all elements of the set are comparable), then the ordering is a total ordering (intuitively, the elements can be arranged along a straight line); otherwise it is a partial ordering. A well-ordering is an ordering such that every non-empty subset of the set contains a minimal element, that is, some element m such that there is no x ≠ m in the set such that x < m. A well-ordering on a set A is a linear ordering with the property that every nonempty subset of A has a minimal element.