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.
TopIntroduction
One of the most recent concepts in neutrosophic theory is NeutroAlgebra. Classical algebras consist of a set of elements, one or more internal operations over these elements, and a set of axioms that must be fulfilled for 100% of them. Meanwhile, NeutroAlgebras change this last condition to the one that establishes that at least one axiom or internal operation is satisfied by a percentage other than 0% and 100% of the elements, (Smarandache, 2020a, 2020b).
This theory includes a more general concept called Neutrosophication such that a concept, theory, structure, among others, denoted by <A> can be divided into a triad <A>, <NeutA>, and <AntiA>; where <AntiA> is the opposite of <A>, while <NeutA> is neither <A> nor <AntiA>. Considering the algebras, if <A> represents an algebra (group, ring, skew-field, field) then <AntiA> represents the structure where one of the axioms or an internal law is not satisfied for 100% of the elements, while <NeutA> is not an <A> nor an <AntiA> because in at least one axiom or an internal operation there are strictly less than 100% of elements that satisfy this or those conditions.
Later, F. Smarandache defined NeutroGeometries from Neutroalgebras, (Smarandache, 2021). This is because Geometries can be seen as a special type of algebras, due to they consist of elements (points, lines, planes, hyperplanes) and axioms that must be fulfilled in 100% of the cases. The most classical example is the Euclidean geometry, where a series of axioms hold. The most controversial of them is the fifth Euclid’s postulate wherein its most popular version says that given a straight line l and a point P outside it, one and only one line parallel to l passes through P. However, in NeutroGeometry at least one axiom is fulfilled by less than 100% of the elements and in more than 0% at the same time. Likewise, Neutrosophication is the method to obtain NeutroGeometry from Geometry.
These ideas, although very recent, have a more distant historical antecedent in time with the definition of Smarandache Geometries that consist of geometries that in the same space contain diverse types of classical Geometries, (Iseri, 2002). In the Smarandache Geometry the model of Euclid’s fifth postulate is satisfied for some elements of the space and others do not satisfy it in two different ways, either no parallel passes through the point P, which are the so-called Riemannian Geometries, or there are an infinite number of parallels passing through P, which are called hyperbolic geometries. These models have been of interest to physics because they are a new approach to geometry, where mixed geometries generalize and at the same time model in a more truthful way the behavior of physical spaces that is the basis of Albert Einstein's Theory of Relativity and the theory of parallel universes, (Rabounski, 2010).
However, NeutroGeometries can be extended to other branches of geometry that are of interest to mathematics and its applications. This chapter deals with finite geometries, which consist of two types of elements, the points and the lines, whose sets are of finite cardinality, (Dembowski, 1997; Kiss & Szonyi, 2020). Within finite geometries, two types of them are studied: projective geometries and affine geometries. None of them is Euclidean because they are not infinite.
Projective spaces and especially finite projective planes do not contain parallels, since one of its axioms specifies that any two lines intersect at a point. Whereas finite affine geometries contain bundles of lines that form a parallelism equivalence class. Both geometries are related to each other since an affine space can be obtained from a projective space and vice versa, (Dissett, 2000; Hirschfeld, 1979).
These geometries can be represented by combinatorial mathematics or linear algebra. Both, projective and affine geometries are based on incidence structures, which are relations where if the point P passes through the line l, it is said that P is incident with l and also l is incident with P. As mentioned above, although projective and affine spaces can be obtained each one from the other, the axiom related to the parallelism of projective spaces is contradictory with its similar axiom in affine spaces.
Key Terms in this Chapter
NeutroGeometry: A geometric structure whose definition or at least one axiom is simultaneously satisfied by some elements and not satisfied by others.
Finite Geometry: Geometric system with only a finite number of points.
Finite Field: Algebraic structure consisting of a finite set of elements and two internal operations + and ×, such that + and × are commutative groups except for 0 in ×; 0 is the neutral element of +; 1 is the neutral element of ×; + and × are distributive each other; 0 has no inverse for ×.
Finite Vector Space: Vector space over a finite field.
Principle of Duality: Principle satisfied by the projective spaces where the theorems which link some geometric elements are also valid if they are interchanged. E.g., in the finite projective plane every theorem which links the terms points, lines, and the incidence is valid where the two firsts of them are interchanged.
Projective Geometry: Geometric structure where there is non-parallelism of the lines in the same plane. Moreover, every pair of planes meets in a line, and so on.
Incidence Matrix: A matrix whose every row represents every line of the finite plane and every column represents every point, or vice versa, such that the element of the matrix is 1 if there is an incidence relation between the corresponding pair of line and point, and it is 0 otherwise.
Incidence Structure: Symmetric relation among points and lines, where if a line l contains a point P it is said that l is incident with P and P is incident with l .
Mixed Geometry: Also called Smarandache geometry , it is the geometry where some elements satisfy the axiomatic of some type of geometry, and other elements satisfy other types of geometries, all in the same space. E.g., there are geometric models which are partially Euclidean, partially hyperbolic, and partially elliptic.
Affine Geometry: Geometric structure such that given a line and an external point, there is a unique line parallel to the aforementioned one, which contains the point.