ABSTRACT
In Chapters 11 and 16 through 18, we see that there are hyper-reguli which are not Andre´ hyper-reguli but nevertheless are built from Andre´ nets; the replacements not being Andre´ replacement.
So, the question arises whether there are hyper-reguli of order qn and degree (qn−1)/(q−1), which are not in fact Andre´ hyper-reguli, even though we know that not all replacements of Andre´ hyper-reguli need to lead to Andre´ planes. When n = 3, it is known by Bruck [178] that every hyper-regulus is an Andre´ hyperregulus. However, it still is possible that a set of Andre´ hyper-reguli might not correspond to an Andre´ set of hyper-reguli; that is, might not determine an Andre´ plane (in the latter case, the set is said to be ‘linear’). In fact, this is done in Dover [323, 325], where sets of two and three Andre´ hyper-reguli are constructed that are not linear, thus constructing non-Andre´ translation planes. Furthermore, Culbert and Ebert [267] have determined several classes of sets of hyper-reguli of order q3, of large order. In addition, Basile and Brutti [103, 104] analyze sets of hyper-reguli of order q3 that produce either Andre´ or generalized Andre´ planes.
