Inequivalent quantizations from gradings and Z2 x Z2 parabosons

This paper introduces the parastatistics induced by Z(2) x Z(2)-graded algebras. It accommodates four kinds of particles: ordinary bosons and three types of parabosons which mutually anticommute when belonging to different type (so far, in the literature, only parastatistics induced by Z(2) x Z(2)-graded superalgebras and producing parafermions have been considered). It is shown how to detect Z(2) x Z(2)-graded parabosons in the multi-particle sector of a quantum model. The difference with respect to a system composed by ordinary bosons is spotted by measuring some selected observables on certain given eigenstates. The construction of the multi-particle states is made through the appropriate braided tensor product. The application of Z(2)- and Z(2) x Z(2)-gradings produces 9 inequivalent multi-particle Hilbert spaces of a 4 x 4 matrix oscillator. The Z(2) x Z(2)-graded parabosonic Hilbert space is one of them.