Bachelor latin squares with large indivisible plexes

In a latin square of order n, a k-plex is a selection of kn entries in which each row, column, and symbol occurs k times. A 1-plex is also called a transversal. A k-plex is indivisible if it contains no c-plex for 0= 4, there exists a latin square of order n that can be partitioned into an indivisible left perpendicular n/2 right perpendicular-plex and a disjoint indivisible left perpendicular n/2 right perpendicular-plex. For all n >= 3, we prove that there exists a latin square of order n with two disjoint indivisible left perpendicular n/2 right perpendicular-plexes. We also give a short new proof that, for all odd n >= 5, there exists a latin square of order n with at least one entry not in any transversal. Such latin squares have no orthogonal mate