On the lattice of $ES_I$-closed classes of multifunctions on two-elements set

The paper considers multifunctions on a two-element set with superposition and the equality predicate branching operator. The superposition operator is based on the intersection of sets. The main purpose of the work is to describe all closed classes with respect to the considered operators. The equality predicate branching operator allows the task to be reduced to a description of all closed classes generated by 2-variable multifunctions. Using this, it is shown that the lattice of classes closed with respect to the considered operators contains 237 elements. A generating set is specified for each closed class. The result obtained in the paper extends the known result for all closed classes of partial functions on a two-element set.