Inequivalent quantizations from gradings and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ parabosons

This paper introduces the parastatistics induced by ${\mathbb Z}_2\times {\mathbb Z}_2$-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 ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded superalgebras and producing parafermions have been considered). It is shown how to detect ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-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 ${\mathbb Z}_2$-and${\mathbb Z}_2\times {\mathbb Z}_2$-gradingsproduces$9$inequivalentmulti-particleHilbertspacesofa$4\times 4$ matrix oscillator. The ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded parabosonic Hilbert space is one of them.