$ES_I $-closure of rank $2$ multi-functions: completeness criterion, classification, and types of bases

Multifunctions are functions that are defined on a finite set and return subsets of the considered set as their values. This paper deals with the closure of multifunctions that can be obtained using the operations of adding dummy variables, composition operator, and operator with the equality predicate branching. We obtain nine precomplete closed classes of multifunctions of rank two and prove the completeness criterion. A classification of multifunctions based on belonging to precomplete classes is presented, and all types of bases are described.