Chapter 4.2, Problem 34E

Suppose that $\left(A_1,{\underset{\_}{\prec }}_1\right)$ and $\left(A_2,{\underset{\_}{\prec }}_2\right)$ are partial orders.

1. Show that the definition $\left(x_1,x_2\right)\underset{\_}{\prec }\left(y_1,y_2\right)$ if and only if $x_1{\underset{\_}{\prec }}_1{y}_1$ and $x_2{\underset{\_}{\prec }}_2{y}_2$ for $\left(x_1,x_2\right),\left(y_1,y_2\right)\in A_1\times A_2$ makes $A_1\times A_2$ a partially ordered set.

2. Let $A_1=A_2=\left\{2,3,4\right\}$. Assign to $A_1$ the partial order $\le$ and to $A_2$ the partial order $|$. Partially order $A_1\times A_2$ as in part (a). Show all relationships of the form $\left(x_1,x_2\right)\underset{\ne }{\prec }\left(y_1,y_2\right)$.

3. Draw the Hasse diagram for the partial order in (b).

4. Find any maximal, minimal, maximum, and minimum elements that may exist in the partial order of (b).

5. With $A_1$ and $A_2$ as in part (b), find the glbs and lubs that exist for each of the following pair of elements.

   1. $\left(2,2\right),\left(3,3\right)$

   2. $\left(4,2\right),\left(3,4\right)$

   3. $\left(3,2\right),\left(2,4\right)$

   4. $\left(3,2\right),\left(3,4\right)$

6. Show, by example that if $\left(A_1,{\underset{\_}{\prec }}_1\right)$ and $\left(A_2,{\underset{\_}{\prec }}_2\right)$ are total orders then $\left(A_1\times A_2,\underset{\_}{\prec }\right)$ need not be a total order.