Abstract
We analyse closed classes in k-valued logics containing all linear functions modulo k. The classes are determined by divisors d of a number k and canonical formulas for functions. We construct the lattice of all such classes for k = p2, where p is a prime, and construct fragments of the lattice for other composite k.
Originally published in Diskretnaya Matematika (2021) 33, №2, 100–116 (in Russian).
Acknowledgment
The author expresses his deep gratitude to V. B. Alekseev for his careful reading of the manuscript and for many useful suggestions.
References
[1] Yablonskii S. V., “Functional constructions in a k-valued logic”, Tr. MIAN SSSR, 51, Izd-vo AN SSSR, Moscow, 1958, 5–142.Search in Google Scholar
[2] Cherepov A.N., “Description of the structure of closed classes in pk containing a class of polynomials”, Problemy Kibernetiki, 1983, №40, 5-18 (in Russian).Search in Google Scholar
[3] Cherepov A. N., “Superstructure of the Class of k-Valued Logic Functions that Preserve Congruences with Respect to All Divisors of k”, Extended Abstract of Candidate’s Dissertation in Mathematics and Physics, 1986.Search in Google Scholar
[4] Meshchaninov D. G., “Some conditions for the representability of functions from Pk by polynomials modulo k”, Dokl. Math., 37:2 (1988), 338–341.Search in Google Scholar
[5] Meshchaninov D. G., “Superstructures of the class of polynomials in Pk”, Math. Notes, 44:5 (1988), 950–954.10.1007/BF01158427Search in Google Scholar
[6] Remizov A. B., “Superstructure of the closed class of polynomials modulo k”, Discrete Math. Appl., 1:1 (1991), 9–22.10.1515/dma.1991.1.1.9Search in Google Scholar
[7] Meshchaninov D. G., “Onthesecond p-differences of functions of pα-valued logic”, Discrete Math. Appl., 3:6 (1993), 611–621.10.1515/dma.1993.3.6.611Search in Google Scholar
[8] Gavrilov G. P., “On the superstructure of the class of polynomials in multivalued logics”, Discrete Math. Appl., 6:4 (1996), 405–412.10.1515/dma.1996.6.4.405Search in Google Scholar
[9] Gavrilov G. P., “On the closed classes of multivalued logic containing the polynomial class”, Discrete Math. Appl., 7:3 (1997), 231–242.10.1515/dma.1997.7.3.231Search in Google Scholar
[10] Krokhin A. A., Safin K. L., Sukhanov E. V., “On the structure of the lattice of closed classes of polynomials”, Discrete Math. Appl., 7:2 (1997), 131–146.10.1515/dma.1997.7.2.131Search in Google Scholar
[11] Meshchaninov D. G., “On the first d-differences of functions of k-valued logic”, Matematicheskie problemy kibernetiki, 7, Nauka, Moscow, 1998, 265–280 (in Russian).Search in Google Scholar
[12] Meshchaninov D. G., “On closed classes of k-valued functions preserving the first d-differences”, Matem. Voprosy Kibernetiki, 8, Nauka, Moscow, 1999, 219–230 (in Russian).Search in Google Scholar
[13] Zaets M. V., “Functions with variative-coordinate polynomiality over primary rings of residues”, Prikl. Diskr. Mat., 2014, №3(25), 12–27 (in Russian).10.17223/20710410/25/2Search in Google Scholar
[14] Zaets M. V., “Classification of the functions over primary ring of residues considered in connection with the method of coordinate linearization”, Prikl. Diskr. Mat. Suppl., 2014, №7, 16–19 (in Russian).Search in Google Scholar
[15] Meshchaninov D. G., “Closed classes of polynomials modulo p2”; Discrete Math. Appl., 28:3 (2018), 167–178.10.1515/dma-2018-0016Search in Google Scholar
[16] Meshchaninov D. G., “A family of closed classes in k-valued logic”, Moscow Univ. Comput. Math. Cybern, 43 (2019), 25–31.10.3103/S0278641919010059Search in Google Scholar
[17] Meshchaninov D. G., “Families of closed classes in pk defined by additive and polynomial representations of functions”, Proc. XII Internat. O. B. Lupanov’s Seminar “Discrete Mathematics and Its Applications” (Moscow, MSU, June 20–25, 2016 ), Publ. Dep. Mech. Math. MSU, Moscow, 2016, 96–106.Search in Google Scholar
[18] Meshchaninov D. G., “Classification of k-valued functions using additive formulas”, Proc. International workshop “Syntax and semantics of logical systems” (August 11–16, 2019, Camp site on the shore of Lake Hovsgol, Mongolia), IGU Publ., Irkutsk, 2019, 68–72.Search in Google Scholar
[19] Gavrilov G. P., Sapozhenko A. A., Problems and exercises in discrete mathematics, Kluwer Acad. Publ., Dordrecht, 1996, xi+422 pp.10.1007/978-94-017-2770-9Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The site-perimeter of compositions
- Some cardinality estimates for the set of correlation-immune Boolean functions
- Continuality of classes of functions in multivalued logic with minimal logarithmic growth rate
- Completeness criterion with respect to the enumeration closure operator in the three-valued logic
- Some families of closed classes in Pk defined by additive formulas
- Finding periods of Zhegalkin polynomials
- Admissible and Bayes decisions with fuzzy-valued losses
Articles in the same Issue
- Frontmatter
- The site-perimeter of compositions
- Some cardinality estimates for the set of correlation-immune Boolean functions
- Continuality of classes of functions in multivalued logic with minimal logarithmic growth rate
- Completeness criterion with respect to the enumeration closure operator in the three-valued logic
- Some families of closed classes in Pk defined by additive formulas
- Finding periods of Zhegalkin polynomials
- Admissible and Bayes decisions with fuzzy-valued losses