Combined Orders
Why
Given multiple orders, can we combine them?
Discussion
Suppose we have two orders $\prec_1$ and $\prec_2$ on $A$. Define $\prec$ by $a \prec b$ if and only if $a \prec_1 b$ and $b \prec_2 a$. Notice that $\prec$ is reflexive, transitive, and antisymmetric, and so it is a partial order. Call it the combined order.
Here's the rub. Even if $\prec_1$ and $\prec_2$ are total, $\prec$ may not be total.1 Consider the basic case $A = \set{a, b}$ and $\prec_1 \; = \set{(a,a), (a, b), (b,b)}$ and $\prec_2 \; = \set{(a,a), (b,a), (b,b)}$ Then $\prec \; = \set{(a,a), (b,b)}$, a partial order, to be sure, but not really any order at all.
There is not anything to be done about it, it is a fact. Total orders do not (necessarily) induce total combined orders.
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