A partial order ≤ on a set X is a total order if it is connected; that is, for all x,y ∊ X either x ≤ y or y ≤ x. Well-orders are total orders; the usual order on the reals is a total order without being a well-order; divides on the positive integers is not a total order as neither of 6 and 9 divides the other.