### Classifications of Boolean functions and their closed sets

Maydim Malkov
2014 SOP Transactions on Applied Mathematics
We have continued and completed Post's classification of two-valued functions and their closed sets. We use minimal bases of closed sets and two Mal 0 cev's algebras instead of preserved relations and universal algebra. Then we use natural classifications of functions that are objects of the first Mal 0 cev's algebra, and closed sets of functions that are objects of the second Mal 0 cev's algebra. Classes of the classifications are disjoint. All these allow us to find more deep properties of
more » ... ctions and, in particular, to find fictitious closed sets that have gross volume and that give big gaps in Post's classification. These allow also to get completeness of classification of closed sets. Post's classification makes up 22% of the complete classification. Our class is a part of a closed set, the part is the rest of a closed set after removal all other closed sets in it. We call a closed set fictitious, if it becomes empty after removing all other closed sets containing in it. So fictitious closed sets are useless for classification of functions. But they take part in classification of closed sets since any classification must contain all objects of a theory. It turns out that non-fictitious closed sets have one-membered bases. These closed sets form level 1 in the multi-level classification of closed sets. Level i contains only closed sets without (i 1)-membered basis but with i-membered basis. We prove that levels 4 and above are empty. The classification by membered bases is well for many-valued functions, too. 1 The survey [4] did not mention any Mal 0 cev's works but contained some results of his works. In [5] Ivo Rosenberg gave Mal 0 cev's results and marked "Preiterative sets are slightly more general than clones". In reality, the preiterative algebra more general than the algebra of clones (Rosenberg used "sets" instead of "algebra"). The monograph [6] did not mention the Rosenberg work. It mentioned many of Mal 0 cev's works but included Mal 0 cev's iterative algebra and excluded Mal 0 cev preiterative algebra. In 2.6 we prove that iterative algebra is not sound. 2 Almost all contemporary researches use unnatural classifications of closed sets of two-and many-valued functions and, as a rule, use not complete classifications 3 The generally accepted name of the main set is E 2 , sometimes A 2 . We use N 2 since it is the subset of set N of natural numbers.