On certain finite geometric structures

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 (Formula presented.) and (Formula presented.) where (Formula presented.) for all (Formula presented.) then A is a semifield if the following holds true: 1. (Formula presented.) is a group and (Formula presented.) 2. The identity map (Formula presented.) 3. If (Formula presented.) then there exists a unique (Formula presented.) such that (Formula presented.), and conversely if (Formula presented.) is a vector space over (Formula presented.) and (Formula presented.) satisfying (Formula presented.) and (Formula presented.) then (Formula presented.) is a semifield. • If T is the group of translations of the affine plane Π and (Formula presented.) is the group of elations of the projective plane (Formula presented.) with axis (Formula presented.) then 1. (Formula presented.) and 2. (Formula presented.).