Sokhatsky F. Associates and decompositions of multiary operations

The thesis is devoted to the study of multiary operations with respect to their composition. The theory of associates and polyagroups are given: it is proved that every partial associative groupoid being surjective and injective is an associate; the full decomposition of an associate operation having a partial invertible element are found, an up to isomorphism description of polyagroups modulo groups is given; a dependence between the classes of polyagroups and groups are found. The well-known problems for polyagroups connecting with the skew operation (when it is: constant, an endomorphism of the polyagroup, regular and has a finite order) are solved. It is proved that every reducible IP-loop is trivial. The notions of cross isotopy and cross isomorphisms of operations are introduced and investigated. A tool for classification of functional equations on quasigroup operations has been developed. The obtained results are applied to the study of identities on quasigroups. A system of associativity functional equations is solved on a set of operations having an invertible element. A new method of abstruct characterization founding of operation algebras is given.