Some topics of Geometry

Abstract

• Several branches of geometry can be treated as certain kinds of hypergroups, known as join spaces. Introduced by W. Prenowitz and studied afterwards by him together with J. Jantosciak, the concept of a join space is “sufficiently general to cover the theories of ordered and partially ordered linear, spherical and projective geometries, as well as abelian groups”.

• If we consider a spherical geometry and identify antipodal points, we obtain a projective geometry. This construction can be described in the context of join spaces as follows:

Let J be the set of points of a spherical join space and for any a ∈ J, let $\bar a = \{ a,{a^{ - 1}}\}$ .Let $\bar J = \{ \bar a|a \in J\}$ . We define on $\bar J$ the following hyperoperation:

$$ \bar ao\bar b = \{ \bar x \in \bar a \cdot \bar b\} $$

, where is the hyperoperation of the spherical join space. Theorem. (see [168]) $(\bar J,o)$ is a projective join space, such that $\forall \bar a \in \bar J,\bar ao\bar a = \bar a/\bar a = \{ \bar e,\bar a\}$, where ë is the identity.