Classification of Closed Sets of Functions in Multi-valued Logic

Using Mal’cev’s preiterative algebra (iterative algebra is not sound) we construct in the first time the natural classification of closed sets of functions in multi-valued logic: every closed set belongs only one class and the classes are disjoint. All contemporary papers construct only intersecting classes. We confirm Post’s thesis that multi-valued logic does not contain anything special in compare with 2-valued logic. There are non-fictitious and fictitious closed sets. The number of non-fictitious closed sets is countable and the number of fictitious closed sets is continuum. This is natural since number of fictitious objects in every theory is very much in comparison with non-fictitious objects. Fictitious closed sets are useless for classification of functions of logic like fictitious variables are useless for calculation of function. Non-fictitious closed sets can have singleton bases. Any fictitious closed set is a join of non-fictitious closed sets and has only multi-membered basis. This is used for classification of closed sets. The class U0 contains closed sets without basis. A class Um does not contain closed sets with (m− 1)-membered bases but contains closed sets with m-membered bases. All the classes are countable. The class Uω contains closed sets with infinite-membered bases. The class is continuum. We construct only the upper level of classification since fictitious closed sets need not have more levels. Only class U1 of non-fictitious closed sets must have more detail classification. But contemporary papers deal with useless fictitious closed sets instead of very important non-fictitious.