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

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We will present 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 PBF⁽¹,¹⁾, we will study a class of Fock--like representations of this algebra, dependent on a positive integer 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 Klein group gradings (for both the carrier spaces and the algebra itself) will be established.