$E$-closed sets of hyperfunctions on two-element set

Hyperfunctions are functions that are defined on a finite set and return all non-empty subsets of the considered set as their values. This paper deals with the classification of hyperfunctions on a two-element set. We consider the composition and the closure operator with the equality predicate branching ($E$-operator). $E$-closed sets of hyperfunctions are sets that are obtained using the operations of adding dummy variables, identifying variables, composition, and $E$-operator. It is shown that the considered classification leads to a finite set of closed classes. The paper presents all 78 $E$-closed classes of hyperfunctions, among which there are 28 pairs of dual classes and 22 self-dual classes. The inclusion diagram of the $E$-closed classes is constructed, and for each class its generating system is obtained.