ABSTRACT
In Chapter 10, we gave a construction of partial t-spreads starting from Desarguesian partial left t-parallelisms, which would construct …rst the putative left subplanes of a right pseudo-regulus net. In this chapter, we consider, conversely, a construction that gives the right pseudo-regulus net directly. The main problem with reversing these ideas to construct the right pseudo-regulus net directly involves taking a t-spread of a vector space of dimension zt over a skew…eld K and forming the ‘direct product’to hopefully obtain a net. However, if z is not 2; then there will be a translation Sperner space constructed from a t-spread by taking translates of the t-subspaces of the spread, which is not a net. These ideas will be revisited and considered in a more general manner when we discuss general spreads and the a¢ ne structures that they generate. In any case, for this chapter, we will consider t-spreads in vector spaces of dimension 2t over skew…elds K. This will allow that any t-spread will de…ne a translation plane. The direct product of two nets is also a net, and so the direct product of two a¢ ne translation planes will also be a net. This will allow the consideration of the arbitrary product of a¢ ne translation planes thus constructing various nets. Many of these ideas have been considered in the Subplanes text
[114], when t = 2 and the material there considers the direct product of two isomorphic translation planes (the 2-fold product). Here we consider a more general theory of n-fold products of n isomorphic translation planes. Some of the early material in this text is also in the Subplanes text. There are a variety of ways that one might de-…ne a ‘direct sum’or ‘direct product’of point-line geometries. In our chapters on Sperner spaces and focal-spreads and their construction, we give other de…nitions of direct sums. So, the reader might be aware that there could more than one meaning for these terms. We …rst shall consider the direct product of two a¢ ne planes. This
may be generalized to more general products of nets and other structures.
