Multiplicative posets

A functionf from the posetP to the posetQ is a strict morphism if for allx, y ? P withx<y we havef(x)<f(y). If there is such a strict morphism fromP toQ we writeP ? Q, otherwise we writeP $$\not \to $$ Q. We say a posetM is multiplicative if for any posetsP, Q withP $$\not \to $$ M andQ $$\not \to $$ M we haveP ×Q $$\not \to $$ M. (Here (p₁,q₁)<(p₂,q₂) if and only ifp₁<p₂ andq₁<q₂.) This paper proves that well-founded trees with height =? are multiplicative posets.