Boolean concept lattices are fundamental structures in formal concept analysis, both from a theoretical and an applied point of view. There are multiple ways to generalise them in the triadic concept analysis framework and one of them, the so-called powerset trilattice, has already been proposed by Biedermann in 1998. However, it lacks some interesting properties such as extremality in the number of triconcepts for tricontexts of a given size. In this paper, we discuss another generalisation of Boolean concept lattices that exhibit such properties. We argue that those structures form equivalence classes and should be studied as such, and investigate the minimum number of objects required to produce them.