 Comparator composition

To combine more and less significant digital comparators, compute $E_H=\neg (L_H\vee G_H)$ and use the displayed formulas. Unequal upper blocks decide the order, because their difference outweighs every possible lower-block difference; equal upper blocks delegate the decision to the lower block. This is an associative operation on the three comparison outcomes. A balanced binary tree of these combinations has $O(n)$ circuit size and $O(\log n)$ circuit depth for $n$-bit inputs.