Introduction to the Finite NeutroGeometries

Abstract

NeutroGeometries generalize geometries in the same way that NeutroAlgebras generalize universal and partial algebras. NeutroGeometry is not one kind of classical geometry, but it can be a combination of some of them in the same space. For the first time, this chapter introduces notions of NeutroGeometry for finite geometries. Finite geometries consist of incidence structures where the set of points has finite cardinality. Usually, they are either projective or affine geometries. In this chapter, the author is mainly focused on the definition of the mixed projective-affine geometry (MPA geometry), which is a NeutroGeometry that follows the line of Smarandache's ideas of defining mixed geometries. The properties of the MPA spaces are studied, especially the one related to the satisfaction of parallelism for some lines and the no satisfaction for others. Additionally, other approaches to this theory are introduced, where elements of Neutrosophy are combined with the incidence matrices obtained from finite geometries. The chapter explores the advantages to use MPA in cryptography.