A Hyperbolic Non Distributive Algebra in 1 + 2 Dimensions

On the Geometry of the Hyperbolic Scator Space in 1+2 Dimensions

“…To begin with, we reconsider some main points in reasoning leading to physical interpretation of scators, as introduced in [1]. The idea of describing physical phenomena along lines of the scator framework takes its origin in the fact that multiplication of scators mimics the shape of velocity addition formula known from special relativity (SR).…”

“…By the way, the scator appearing on the right-hand side of (1.5) will be referred to as dual to o c (see Definition 2.1 below). Many properties of the scator product (1.2) were widely investigated in many contexts [1,8,9], also physical [4]. To gain some insight in possible physical interpretation of the scator algebra, we recall here some basic terminology from [1] The classification proposed above is analogous to what is well known from special relativity.…”

“…The first three elements span the scator space S. The basis {1 1 1, i i i 1 , i i i 2 } is related to {1,ê 1 ,ê 2 } used in papers [1,8]. As in these papers, we demand that…”

“…Following Fernández-Guasti and Zaldívar [1] we consider a commutative, nondistributive 1 + 2 dimensional algebra S, which is also associative provided that divisors of zero are excluded. The elements of this algebra will be called 1 + 2 dimensional scators [1].…”

“…The elements of this algebra will be called 1 + 2 dimensional scators [1]. Scators are denoted by o a = (a 0 ; a 1 , a 2 ), where components a 1 , a 2 are referred to as director components, and a 0 is usually called scalar component, or, in physical context, temporal component.…”