Classification of multioperations of rank $2$ by $E$-precomplete sets

In this paper multioperations defined on a two-element set and their closure operator based on composition operator and the equality predicate branching operator is considered. The composition operator is based on union of sets. The classification of multioperations based on their membership in precomplete sets has been obtained. It is shown that the number of equivalence classes is $129$. All types of bases are described and it is proved that the maximum cardinality of a basis is $4$.