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 E Q such that |N | is odd and Q/N ∼= Z2m for m > 1, and (C) P (Q) intersects the associator subloop of Q. We prove (A) =⇒ (C) and (B) ⇐⇒ (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) =⇒ (A)). We also establish properties of P (Q), prove a generalization of the Dénes-Hermann theorem, and present an elementary proof of a weak form of the Hall-Paige conjecture.