A partial ordering on a set is a relation < that is transitive and reflexive and antisymmetric. That is,

1. (i) x < y & y < z → x < z;

2. (ii) x < x;

3. (iii) x < y & y < x → x=y. If we add

4. (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 π‎ 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.