The algebras of inhibited parafermionic violation of the Pauli principle

The theory of the algebras of Ignat’ev and Kuz’min and of Greenberg and Mohapatra is developed. These are algebras of inhibited parafermionic oscillators that describe possible small violations of the Pauli exclusion principle. The cyclic modules of the algebras are constructed so as to exhibit the connection of their fundamental properties, as uniqueness and statistical signature, with the relations in the algebras. The formalism is presented in such a way suited for further mathematical and physical developments, and has a natural continuation in a forthcoming paper.