# Thread: Order Relations and Predecessors

1. ## Order Relations and Predecessors

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.

2. Originally Posted by tn11631
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.
(1)

All elements have a predecessor except for (x,1) for all positive integers x. Smallest element is (1,1).