On certain finite geometric structures

Abstract

The purpose of this article is to investigate certain finite geometric structures, in particular semifields over finite fields, spread-sets, translation planes as a special type of affine planes, and projective planes. Quasifields and their normed spread-sets are also considered. Among other results, the following main results are proved. • If A is a vector semifield a finite field and where for all then A is a semifield if the following holds true: 1. is a group and 2. The identity map 3. If then there exists a unique such that , and conversely if is a vector space over and satisfying and then is a semifield. • If T is the group of translations of the affine plane Π and is the group of elations of the projective plane with axis then 1. and 2.