We study a class of diagram geometries, achieve a characterization of extended dual affine planes, and embed extended dual affine planes in extended projective planes. The geometries studied are rank 3 diagram geometries such that the residue of a point is a dual net, and the residue of a plane is linear; the dual of such a geometry has partitions on lines and planes which are reminiscent of parallelism of lines and planes of an affine 3-space. Examples of these geometries (some in dual form) include extended dual affine planes, Laguerre planes, 3-nets, and orthogonal arrays of strength 3. Theorem: Any such finite geometry satisfying Buekenhout's intersection property, and such that any two points are coplanar, is an extended dual affine plane (and has order 2, 4, or 10). Theorem: This geometry may be embedded in an extended projective plane of the same order.
NOTATION AND BACKGROUND
An
incidence structure p = (P, L), where elements of P and L are called points and lines respectively, is semilinear (linear) if any two points are on at most one line (exactly one line), p isfirm if P w L :# 0 and each point (line) is on at least two lines (points). A net is an incidence structure in which the set of lines is partitioned into at least two classes (called parallel classes) so that (i) the lines of each parallel class partition the point set, and (ii) lines in distinct classes intersect. If a net has at least three parallel classes of lines, then it is known that every line has the same number of points. We follow Buekenhout's revised foundation [2] of the topic of diagrams for geometries. When we use the phrase 'a geometry admitting the diagram D' we mean, in the language of [2], 'a firm, strongly connected geometry belonging to diagram D'.
We present our dictionary of rank 2 diagrams. We will say that a firm connected incidence structure p admits diagram zr L c net Af 0 O, O O, O O.