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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.

Multioperations are operations from a finite set A to set of all subsets of A. The usual composition operator leads to a continuum of closed sets. Therefore, the research of closure operators, which contain composition and other operations becomes necessary. In the paper, the closure of multioperations that can be obtained using the operations of adding dummy variables, identifying variables, composition operator, and operator with the equality predicate branching is studied. We obtain eleven precomplete closed classes of multioperations of rank 2 and prove the completeness criterion. The diagram of inclusions for one of the precomplete class is presented.

The construction of Galois theory for the algebras of operations and relations is a popular topic for investigation. It finds numerous applications in both algebra and discrete mathematics – especially for the perfect Galois connection, since if such a connection is established for the sets of all subalgebras of some algebra then the algebraic closure in this algebra coincides with the Galois closure, and this is an efficient tool for solving many algebraic problems. The perfect Galois connection is well known for clones and co-clones, as well as for some algebras of operations, for example, for clones and superclones. In all these cases, infinite algebras are considered. In this article we study Galois theory for finite algebras of operations and multioperations with fixed rank and arbitrary dimension. We find the necessary and sufficient conditions on the dimension of algebras so that the Galois connection between the lattices of subalgebras of algebras of operations and algebras of multioperations of rank 2 is perfect. The problem of finding the necessary and sufficient conditions for the existence of a perfect Galois connection between lattices of algebras of operations and algebras of multioperations of arbitrary fixed rank is posed.

In algebras of multioperations, unlike algebras of operations, the superassociativity identity does not hold, but only the semi-superassociavity identity is true. For a more detailed study of the identities satisfiable in fixed dimension algebras of multioperations, this work defines the variety to which these algebras belong. In particular, among these identities defining a variety, an identity is introduced that similar to the Dedekind relation for binary relations. From the introduced identities, some consequences are derived that satisfiable in the fixed dimension algebras of multioperations. Note that the variety is defined in a language whose symbols are interpreted by the superposition metaoperations, the first argument permissibility, and constant projection metaoperations for each argument and the zero multioperation. In this language, the terms are the intersection meta-operations, the permissibility by any argument, the full multioperation, and the inclusion multioperation. Another interesting task is studying quasiidentities satisfiable in the fixed dimension algebras of multioperations.