The Boolean lattices are fundamental algebraic structures in Lattice Theory and Mathematical Logic. Since the triadic approach to Formal Concept Analysis gave rise to the triadic generalization of lattices, the trilattices, it is natural to ask for the triadic analogue of Boolean lattices, the Boolean trilattices. The first step in establishing Boolean trilattices is the study of powerset trilattices, which are the the triadic generalization of powerset lattices. In this paper, an order-theoretic characterization of the powerset trilattices as certain B-trilattices is presented. In particular, the finite B-trilattices are (up to isomorphism) just the finite powerset trilattices. They have 3ⁿ elements. Further topics are the triadic de Morgan laws, the cycles of triadic complements as the triadic complementation and the atomic cycles, which take over the role of the atoms in the theory of Boolean lattices.