## Orders of pairs

### Task number: 4150

Which of the following relations on the set \( {\Bbb N} ^ 2 \) (pairs of positive integers) are orders? Which of these orders are linear?

#### Variant

Comparison in both coordinates \(\le_S\):

\((a,b)\le_S(x,y) \Leftrightarrow a\le x \wedge b\le y\)

#### Variant

Comparison in at least one coordinate \(\le_U\):

\((a,b)\le_U(x,y) \Leftrightarrow a\le x \vee b\le y\)

#### Variant

Comparison in both coordinates in distinct directions \(\le_Z\):

\((a,b)\le_Z(x,y) \Leftrightarrow a\le x \wedge b\ge y\)

#### Variant

Lexicographic comparison \(\le_L\):

\((a,b)\le_L(x,y) \Leftrightarrow a < x \vee (a=x \wedge b\le y)\)

#### Variant

Lexicographic-maximum comparison \(\le_M\):

\((a,b)\le_M(x,y) \Leftrightarrow \max(a,b) < \max(x,y) \vee (a,b)\le_L(x,y)\)

#### Variant

Maximum comparison, but ties are treated lexicographically \(\le_N\):

\((a,b)\le_N(x,y) \Leftrightarrow \max(a,b) < \max(x,y) \vee [\max(a,b)=\max(x,y) \wedge (a,b)\le_L(x,y)]\)