ABSTRACT
Here we …rst construct a variety of parallelisms over the reals that are of the de…ciency one type and then show how to use coset switching to construct many interesting parallelisms. As mentioned above, the application of this construction technique
has been applied most successfully when the spreads other than the Pappian spread are derived conical ‡ock spreads and when the group contains a large normal subgroup that is a central collineation group. (By ‘conical ‡ock spreads,’we intend to mean those spreads that correspond to ‡ocks of quadratic cones.) For example, using Theorems 205 and 206. So, it might be asked if the above constructions can be considered
over in…nite …elds? Here we begin with the consideration of the question when the …eld is the …eld of real numbers, and we are able to show that there are a vast number of parallelisms, depending on the class of strictly increasing functions f on the reals that de…ne a class of conical ‡ock spreads. We point out that our construction process constructs not only parallelisms but (proper) maximal partial parallelisms and actually forms the …rst known classes of such objects. So, we work over the …eld of real numbers K = R, but many of our
arguments will work for arbitrary ordered …elds and we take up a more general analysis later in this chapter. We consider a Pappian spread 1 de…ned as follows:
x = 0; y = x
u t t u
8u; t 2 R.
