Algebras of quasiary and of bi-quasiary relations
Mykola S. Nikitchenko, S.S. Shkilniak · PROBLEMS IN PROGRAMMING · 2016
The notion of quasiary relation which can be considered generalization of the notion of traditional n-ary relation is proposed. A number of algebras of quasiary relations is built and investigated. Alongside with conventional operations of union, intersection, and complement, special nominative operations of renomi-nation and quantification are defined for quasiary relations. The isomorphism between the algebra of quasiary relations and the first-order algebra of total single-valued quasiary predicates is proved. Al-gebras of bi-quasiary relations defined over sets of pairs of quasiary relations are built. The isomorphism between algebras of bi-quasiary relations and alge-bras of quasiary predicates is proved. The following subclasses of algebras of bi-quasiary relations are specified: alge-bras of partial single-valued (functional), total, total many-valued bi-quasiary relations. For all defined subclasses their counterparts of the classes of algebras of quasiary predicates are described. Also subalgebras of the algebra of bi-quasiary relations induced by upward closedness and downward closedness are investigated.