Abstract
We give a classification of dual functions, they are m-al functions. We call a function m-al with respect to an operator if the operator lives any function unchanged after m times of using the operator. And 2 ≤ m ≤ k. Functions with different m have very different properties. We give theoretical results for clones of self-dual (m = 2) and self- -al (m = k) functions in k-valued logic at k ≤ 3. And we give numerical results for clones of self-dual and self-3-al functions in 3-valued logic. In particular, the inclusion graphs of clones of self-dual and of self-3-al functions are not a lattice.
Highlights
Multi-valued logic attracts intensive attention because of the connection with computer technology
We find the numbers of relations preserved by dual and k -al functions
A clone can belong to several clones
Summary
Multi-valued logic attracts intensive attention because of the connection with computer technology. Lau [3], some self-3-al clones were found by S. More complete set of self-3-al clones is given in the article. The set of monotone clones was investigated by Machida [6]. We build the inclusion graphs of clones of self-dual and 3 -al functions and find properties of the functions. The letter k is used for values of logics only. The letter n is used for numbers of variables of functions only. The letter b is used as a value of variables. The letter l is used as values of functions. We use Boolean projections of functions if values of functions are 0 or 1 whenever values of variables are 0 or 1. We use sign “;" to separate values of variables from values of functions in lines of tables of functions.
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