Quasistochastic semigroup as a generalized group

Abstract

Two representations of the hyperbolic-tangent semigroup (HTS) are discussed: the group representation (the Lorentz group) and the semigroup representation (the quasistochastic semigroup of relativistic endomorphisms-QSRE). Through a generalization of the condition for regular conjugation, it is established that the QSRE is, in a certain sense, a generalized group, Invertibility of the representation with respect to zero is shown to be an antiautomorphism (for transforms of fixed points of the HTS). The group representation of the HTS is understood from this point of view to be the invertibility with respect to the trivial unit element. Since the class of forms of HTS idempotents is exhausted by simply the unit and zero representations, there can be no other forms of the principle of relativistic invariance.