Please forgive my ignorance here..
I have the following pairs $cd \succ c\bar{d}$ and $\bar{c}\bar{d} \succ \bar{c}d$ defined over the cartesian product of two variables $C=\{c,\bar{c}\}$ and $D=\{d,\bar{d}\}$ where $\succ$ means better than. This seems unreal question but I am really struggling whether these can be considered as a one order (but with incomparability) or two different total orders. If I wanted to represent them as a one order how to say that the pair $\bar{c}\bar{d}$ is incomparable with $cd$ or $c\bar{d}$?
Either way is fine. In the first case, $\succ$ will be viewed as a partial order on the Cartesian product of the two sets, in the second case: as a total order on each of the respective sets. If you go with the first representation (as one order), you can say that $\bar{c}\bar{d}$ is incomparable with both $cd$ and $c\bar{d}$ as follows: (i) $\bar{c}\bar{d}$ and $cd$ are incomparable because both $\bar{c}\bar{d} \not\succ cd$ and $cd \not\succ \bar{c}\bar{d}$, and (ii) $\bar{c}\bar{d}$ and $c\bar{d}$ are incomparable because both $\bar{c}\bar{d} \not\succ c\bar{d}$ and $c\bar{d} \not\succ \bar{c}\bar{d}$.