Fryz I. Orthogonality of multuary operations and algorithms of their construction.

In the thesis, -ary operations and quasigroups, tuples of operations and their combinatorial properties such as orthogonality and its kinds are studied. The main directions of this work are to study invertibility of an arbitrary composition of two multiary operations; to generalize the recursive algorithm for construction of orthogonal operations; to investigate the dependence between orthogonality of operations and orthogonality of retracts of these operations; to find and investigate the methods for constructing an orthogonal complement of operations and hypercubes and to estimate the number of complements by the obtained algorithms.The perpendicularity concept as a generalization of orthogonality concept of binary operations to a pair of multiary operations of different arities is introduced. Relations between invertibility of a composition of two operations and perpendicularity of some parastrophes of the decomposition components are shown.The algorithm for construction of orthogonal -ary operations that was proposed by G.B. Belyavskaya and G.L. Mullen in 2005 is generalized. Proposed generalization of this algorithm is -block-wise recursive algorithm, one of the parameters of which is some partition of the indices set of variables, and every block of output operations is constructed by a block of new operations and all operations that have been constructed by the previous steps. An algorithm for construction of orthogonal -ary operations using blocks of orthogonal operations of a less arity is suggested.The problem of the dependence between orthogonality of operations and orthogonality of their retracts is studied, namely, it is shown that retract orthogonality implies orthogonality, but the inverse statement is not true. It is proved that every -tuple of orthogonal -ary operations is prolongable to a -tuple of orthogonal -ary operations. The dependence among different generalizations of orthogonality of binary operations (retract orthogonality, strong orthogonality, perpendicularity of maximal type) is described. It is shown that for central quasigroups (linear isotopes of abelian groups) over a prime order field, retract orthogonality is the necessary and sufficient condition for orthogonality.It is well known that every -tuple of orthogonal -ary operations ( ) can be embedded into an -tuple of orthogonal -ary operations, i.e., the existence of an orthogonal complement is proved. In the paper, an algorithm for complementing a -tuple of orthogonal -ary operations to an -tuple of orthogonal -ary operations is found. It is proved that every -tuple of -retractly orthogonal -ary operations is complementable to an -tuple of orthogonal -ary operations by the proposed algorithm. Some estimations of the number of orthogonal complements, in particular, a lower bound and an upper bound of the number of different trivial complements and a lower bound of the number of all possible complements are found.An algorithm for the construction of orthogonal complements of an arbitrary -tuple of orthogonal -ary operations to an -tuple of orthogonal -ary operations for an arbitrary such that is suggested, and a lower bound of the number of such complements is found.A method for constructing an -ary quasigroup having admissible binary retracts is found and a method for constructing a pair of perpendicular quasigroups is described. The algorithms for the constructing and complementing a tuple of orthogonal ternary operations are described and classified into three classes up to parastrophy of defining partitions.