Formally, given two partially ordered sets A and B, the order ≤ on the Cartesian product A × B such that (a,b) ≤ (a′,b′) if and only if a < a′ or (a = a′ and b ≤ b′)
Given sets (A1, A2, ..., An) and their total orderings (1, 2, ..., n), the order d of A1 × A2 × ... × An such that (a1, a2, ..., an) d (b1,b2, ..., bn) iff (∃m > 0) (∀ i i = bi ) and (am m bm )