ABSTRACT
In this chapter, we consider a transversal to a derivable net interpreted in context of the author’s embedding theorem of derivable nets. We recall that the author’s work on derivable nets shows that every derivable net is combinatorially equivalent to a 3-dimensional projective space over a skew…eld K. More precisely, the points and lines of the net become the lines and points skew to a …xed line N . Now consider any a¢ ne plane containing the derivable net and choose any line ` of , which is not a line of the derivable net. This set of points then becomes a set of lines in the combinatorially equivalent structure. For example, if the order of is …nite q2; then this provides q2 lines and adjoining N , we have q2 + 1 lines. Furthermore, Knarr [163] proved that, every line ` not belonging to the derivable net embeds to a set of lines in the projective space such this set union N becomes a spread S(`) of PG(3; K); which is also a dual spread. Now generalize Knarr’s insight but merely assume that we have
a derivable net D that admits a transversal T . Basically, the same ideas will show that in the embedding, we also obtain a spread and hence a corresponding translation plane, but which is not necessarily a dual spread. Now since I was a student of T. G. Ostrom, I had studied what could be said of the geometries that one might obtain from a …nite derivable net that admits a transversal. What happens is that there is a more-or-less direct construction of a dual translation plane. So, on the one hand, Knarr’s ideas give a translation plane from the transversal to a derivable net and Ostrom’s ideas give a dual translation plane. How are the two a¢ ne planes related? The word ‘dual’translation plane might suggest that the two planes are duals of each other when considered as projective planes.
