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.
| Published in | Pure and Applied Mathematics Journal (Volume 6, Issue 2) |
| DOI | 10.11648/j.pamj.20170602.11 |
| Page(s) | 59-70 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2017. Published by Science Publishing Group |
Discreate Mathematics, K-Valued Function Algebra, Selfdual Functions
| [1] | Malkov M. A., Classifications of closed sets of functions in multi-valued logic, SOP transactions on applied math. 1:3, 96-105 (2014), http://www.scipublish.com/journals/AM/papers/download/2102-970.pdf. |
| [2] | Yablonskiy S. V.: Functional constructions in many-valued logics (Russian). Tr. Mat. Inst. Steklova, 515-142 (1958). |
| [3] | Lau D., Functions algebra on finite sets, Springer (2006). |
| [4] | Csàkàny B., All minimal clones on the three-element set, Acta Cybernet., 6, 227-238 (1983). |
| [5] | Marchenkov S. S., Demetrovics J., Hannak L., On closed classes of self-dual functions in P3. (Russian) Metody Diskretn. Anal. 34, 38-73 (1980). |
| [6] | Machida H., On closed sets of three-valued monotone logical functions. In: Colloquia Mathematica Societatis Janos Bolyai 28, Finite Algebra and multiple-valued logic, Szeged (Hungary), 441-467 (1979). |
| [7] | Post E. L., The two-valued iterative systems of mathematical logic. Princeton Univ. Press, Princeton (1941). |
| [8] | Mal’cev A. I., Iterative Post algebras, NGU, Novosibirsk, (Russian) (1976). |
| [9] | Rosenberg, I. G., Űber die Verschiedenheit maximaler Klassen in Pk. Rev. Roumaine Math. Pures Appl. 14, 431–438 (1969). |
| [10] | Malkov M. A. Classifications of Boolean Functions and Their Closed Sets, SOP transactions on applied math. 1:2, 172-193 (2014), http://www.scipublish.com/journals/AM/papers/download/2102-522.pdf. |
APA Style
M. A. Malkov. (2017). Clones of Self-Dual and Self-K-Al Functions in K-valued Logic. Pure and Applied Mathematics Journal, 6(2), 59-70. https://doi.org/10.11648/j.pamj.20170602.11
ACS Style
M. A. Malkov. Clones of Self-Dual and Self-K-Al Functions in K-valued Logic. Pure Appl. Math. J. 2017, 6(2), 59-70. doi: 10.11648/j.pamj.20170602.11
AMA Style
M. A. Malkov. Clones of Self-Dual and Self-K-Al Functions in K-valued Logic. Pure Appl Math J. 2017;6(2):59-70. doi: 10.11648/j.pamj.20170602.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.pamj.20170602.11,
author = {M. A. Malkov},
title = {Clones of Self-Dual and Self-K-Al Functions in K-valued Logic},
journal = {Pure and Applied Mathematics Journal},
volume = {6},
number = {2},
pages = {59-70},
doi = {10.11648/j.pamj.20170602.11},
url = {https://doi.org/10.11648/j.pamj.20170602.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20170602.11},
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.},
year = {2017}
}
TY - JOUR T1 - Clones of Self-Dual and Self-K-Al Functions in K-valued Logic AU - M. A. Malkov Y1 - 2017/03/10 PY - 2017 N1 - https://doi.org/10.11648/j.pamj.20170602.11 DO - 10.11648/j.pamj.20170602.11 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 59 EP - 70 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20170602.11 AB - 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. VL - 6 IS - 2 ER -
Information
Email: