  The thesis is devoted to abstract, geometric and symmetric aspects of modern
elementary particle theories. A new direction in constructing supersymmetric
and superstring models based on consequent and strong consideration and
inclusion of semigroups, ideals and noninvertible properties into their
mathematical structure is proposed. A theory of semisupermanifolds (Chapter I)
and noninvertible generalization of superconformal (Chapters II-III) and
hyperbolic geometries (Chapter V) are introduced. New continuous supermatrix
representations of semigroups (Chapter IV) are obtained. The carried out
investigations will allow us to formulate a theoretical model of elementary
particles based on supersymmetry in terms of more general categories and new
structures - as a theory of abstract and supermatrix semigroups which includes
previous theories as a particular invertible case in abstract sense.
