Let Z+ denote the set of positive integers. Consider the following order relations on Z+ x Z+:

(1) the dictionary order

(2) (x0,y0)<(x1,y1) if either x0-y0<x1-y1, or x0-y0=x1-y1 and y0<y1.

(3) (x0,y0)<(x1,y1) if either x0+y0<x1+y1, or x0+y0=x1+y1 and y0<y1.

Questions: in these order relations, which elements have immediate predecessors? Does the set have a smallest element? Show that all three order types are different.