Does Karp reducibility yield a total order? Or with other words, do we have that for every language $A$ and $B$, $A \leq_p B$ or $B \leq_p A$?
Far from it. Indeed, any countable distributive lattice embeds as a sub-partial-order of $\leq_p$, even if we only consider those degrees in between two given fixed languages (K. Ambos-Spies, Sublattices of the polynomial time degrees, Inform. &amp; Control 65(1):63-84, 1985).