Topological Bihyperbolic Modules

Abstract

The aim of this article is introducing and researching hyperbolic modules, bihyperbolic modules, topological hyperbolic modules and topological bihyperbolic modules. In this regard, we define balanced, convex and absorbing sets in hyperbolic and bihyperbolic modules. In particular, we investigate convex sets in hyperbolic numbers set (it is a hyperbolic module over itself) by considering the isomorphic relation of this set with 2−dimensional Minkowski space. Moreover, bihyperbolic numbers set is a bihyperbolic module over itself, too. So, we define convex sets in this module by considering hypersurfaces of 4−dimensional semi Euclidean space that are isomorphic to some subsets of bihyperbolic numbers set. We also study the interior and closure of some special sets and neighbourhoods of the unit element of the module in the introduced topological bihyperbolic modules. In the light of obtained results, new relationships are presented for idempotent representations in topological bihyperbolic modules.