Products of all elements in a loop and a framework for non-associative analogues of the Hall-Paige conjecture

For a finite loop $Q$, let $P (Q)$ be the set of elements that can be represented as a product containing each element of $Q$ precisely once. Motivated by the recent proof of the Hall-Paige conjecture, we prove several universal implications between the following conditions: (A) $Q$ has a complete mapping, i.e. the multiplication table of $Q$ has a transversal, (B) there is no $N \normal Q$ such that $|N|$ is odd and $Q/N \cong \ZZ_{2^m}$ for $m \geq 1$, and (C) $P(Q)$ intersects the associator subloop of $Q$. We prove $(A) \implies (C)$ and $(B) \iff (C)$ and show that when $Q$ is a group, these conditions reduce to familiar statements related to the Hall-Paige conjecture (which essentially says that in groups $(B) \implies (A))$. We also establish properties of $P(Q)$, prove a generalization of the D\'enes-Hermann theorem, and present an elementary proof of a weak form of the Hall-Paige conjecture.