Products of uniquely completable partial Latin squares (Q2707961)

\textit{R. A. H. Gower} [Critical sets in products of Latin squares, Ars Combinatoria, to appear] conjectures that the completable product of any two partial Latin squares with unique completion also has unique completion. The present authors extend the class of partial Latin squares for which Gower's conjecture is true. A uniquely completable set is strong if one can find a sequence of sets of triples \(U= S_1\subseteq S_2\subseteq\cdots\subseteq S_f= L\) such that each triple \(t\in S_{v+1}\setminus S_v\) is a forced choice. Gower proved that if at least one of \(P\) or \(Q\) is strongly uniquely completable, then the partial Latin square \(P\times Q\) has unique completion. The authors weaken the condition \textit{strong} to \textit{near-strong} and \textit{forced choice} to \textit{semi-forced choice} and prove the same result.