The S-classification of functions of many-valued logic

Abstract

The set of functions of many-valued logic is proposed to be classified with respect to two operations: superposition and transition to dual functions (the ^-classification). The contensive description of all 5-closed classes, which was begun by the author in 1979-82, was completed by Nguen Van Hoa. If fc > 5, then the set of functions of /:-valued logic has only two 5-precomplete classes: the class h of idempotent functions and the Slupecki class SLPk. In this paper the key properties determining the 5-closed classes are found and formalized in the form of the so-called basic relations. Using the Galois theory for Post algebras, it is shown that every 5-closed class of functions, which is not contained in SLPk, can be described by the basic relations. In the set of all systems of the basic relations all independent systems are determined which correspond to all 5-closed classes not contained in SLPk. An exact formula for the number of 5-closed classes contained in Ik is obtained which is a cubic polynomial in k.Let P k be the set of all functions of ^-valued logic [1,2]. Any analysis of functions of many-valued logic is necessarily based on structuring (classification) of subsets of P k . In discrete mathematics the approach based on generative capabilities of functions with respect to superposition has got the most prevalence for classification of subsets of P k . For k = 2 this approach gives the well-known countable Post classification [3,4]. For k > 3 the continuity of arising closed classes [5] is an objective obstacle for constructing and effective using of a similar classification. It compels to seek other, wider, closure operators which would include superposition and permit to avoid familiar undesirable effects.In the papers [6-10] the classifications of functions of many-valued logic self-dual with respect to permutations of the symmetric group and of the alternating group were constructed. The ideas underlying these classifications can be transferred from individual functions to classes of functions. More precisely, operations giving dual functions for permutations of a fixed group should be added to superposition. Above all, it is natural to consider the largest permutation group, that is, the symmetric group, which under this approach leads to the largest closed classes and, therefore, to the least number of closed classes. Several names were offered for the corresponding closed classes and the classification; we call them the 5-closed classes and the 5-classification respectively.

Clone classification of dually discriminator algebras with finite support

“…Dually discriminator algebras play a noticeable role in the theory of universal algebras and multivalued logic, in particular, in the problems of classification of completeness [1,2,[4][5][6][7][8][9]. For instance, nearly a "half' of all homogeneous algebras [10] (including those with infinite supports) are equivalent to appropriate dually discriminator algebras of finite type [2,[5][6][7].…”

“…For instance, nearly a "half' of all homogeneous algebras [10] (including those with infinite supports) are equivalent to appropriate dually discriminator algebras of finite type [2,[5][6][7].…”

On the structure of equationally closed classes

“…(of course, (8) and (10) should obey the necessary condition that the same variables are replaced by the same values from E k ). We thus arrive at a deduction of, generally speaking, another equation '.c 1 ; : : : ; c n / D c. As a whole, all these .c 1 ; : : : ; c n ; c/ are, obviously, values for the vector function …”

“…To reduce the number of variables in vector function (11), for each (8) we identify variables in equation (6) according to the equivalence relation ker defined by (8). We assume that in doing so we use variables x 1 ; : : : ; x k .…”

“…In a series of papers [1][2][3][4][5][6][7][8][9][12][13][14][15][16][17][18][19][20][21], on the set of functions of many-valued logic closure operators are defined, which are much more strong than the usually considered superposition operator. In particular, for any k 2 these operators generate on the set P k of the functions of k-valued logic finite (occasionally, countable) classifications.…”