Let $Z_+$ denote the set of positive integers. Consider the following relations on $Z_+ \times Z_+$:
The dictionary order; that is, $(x_0,y_0) < (x_1,y_1)$ if either $x_0 < x_1$, or $x_0 = x_1$ and $y_0 < y_1$.
$(x_0, y_0) < (x_1,y_1)$ if either $x_0 - y_0 < x_1 - y_1$, or $x_0 - y_0 = x_1 - y_1$ and $y_0 < y_1$.
$(x_0, y_0) < (x_1,y_1)$ if either $x_0 + y_0 < x_1 + y_1$, or $x_0 + y_0 = x_1 + y_1$ and $y_0 < y_1$.
In these order relations, which elements have immediate predecessors?
In which order relations, does the set $Z_+ \times Z_+$ have a smallest element?
How to show that all the above three order types are different?
Let $A$ and $B$ be two non-empty sets with the order relations $<_A$ and $<_B$, respectively. Then the sets $A$ and $B$ are said to have the same order types if there exists a bijective function $f \colon A \to B$ such that $a_1 <_A a_2$ implies $f(a_1) <_B f(a_2)$ for any pair of elements $a_1$, $a_2$ in $A$. Otherwise, $A$ and $B$ are said to have different order types.