Graded Fock-like representations for a system of algebraically interacting paraparticles

We will present and study an algebra describing a mixed paraparticle model, known in the bibliography as The Relative Parabose Set (RPBS). Focusing in the special case of a single parabosonic and a single parafermionic degree of freedom P(1,1)BF, we will construct a class of Fock–like representations of this algebra, dependent on a positive parameter p a kind of generalized parastatistics order. Mathematical properties of the Fock–like modules will be investigated for all values of p and constructions such as ladder operators, irreducibility (for the carrier spaces) and -gradings (for both the carrier spaces and the algebra itself) will be established.