<p> In this thesis we examine a generalized notion of ordinary two dimensional affine and projective geometries The first six chapters deal very generally with coordinatization methods for these geometries and a direct construction of the analytic model for the affine case. The last two chapters are concerned with a discussion of these structures viewed as topological geometries. </p> / Thesis / Doctor of Philosophy (PhD)

We give several characterizations of partition lattices and projective geometries. Most of these characterizations use characteristic polynomials. A geometry is non—splitting if it cannot be expressed as the union of two of its proper flats. A geometry G is upper homogeneous if for all k, k = 1, 2, ... , r(G), and for every pair x, y of flats of rank k, the contraction G/x is isomorphic to the contraction G/y. Given a signed graph, we define a corresponding signed—graphic geometry. We give a characterization of supersolvable signed graphs. Finally, we give the following characterization of non—splitting supersolvable signed-graphic geometries : If a non-splitting supersolvable ternary geometry does not contain the Reid geometry as a subgeometry, then it is signed—graphic.