In this talk we shall discuss the notion of hyperlattices, that is preorders somewhat similar to lattices, but where instead of binary supremums and infimums we consider sets of minimal elements of upper bounds and maximal elements of lower bounds. We will utilize this notion to study superpowersets, that is families of functions defined on nonempty sets with values in hyperlattices, and superpowergroups, that is algebras defined on superpowersets with group-like properties. Numerous examples of such objects appear in topology, logic and algebra, and, in particular, in the algebraic theory of quadratic forms. Finally, we shall distinguish one class of superpowergroups that we call relational superpowergroups, and will see how the category of relational superpowergroups is almost a topos — that is, it satifsies all the axioms of a topos except the subobject classifier axiom. This is joint work with Hugo Mariano and Kaique Matias de Andrade Roberto.