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{\bf Kyle Pula }
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{\bf Products of All Elements in a Loop and a Framework for Non-Associative Analogues of the Hall-Paige Conjecture}
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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:

\item{(A)} $Q$ has a complete mapping, i.e. the multiplication table
of $Q$ has a transversal, 
\item{(B)} there is no $N \trianglelefteq Q$
such that $|N|$ is odd and $Q/N \cong {\Bbb Z}_{2^m}$ for $m \geq 1$,
and 
\item{(C)} $P(Q)$ intersects the associator subloop of $Q$.

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We prove $(A) \Longrightarrow (C)$ and $(B) \Longleftrightarrow (C)$
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 $(B) \Longrightarrow (A))$. 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.

\bye
