Abstract
We show that in multivalued logic there exist a continual family of pairwise incomparable closed sets with minimal logarithmic growth rate and a continual chain of nested closed sets with minimal logarithmic growth rate. As a corollary we prove that any subset-preserving class in multivalued logic contains a continual chain of nested closed sets and a continual family of pairwise incomparable closed sets such that none of the sets is a subset of any other precomplete class.
Note: Originally published in Diskretnaya Matematika (2021) 33,№3, 54–63 (in Russian).
References
[1] Bodnarchuk V. G., Kaluzhnin L. A., Kotov V. N., Romov B. A., “Galois theory for Post algebras. I”, Kibernetika, 1969, №3, 1–10 (in Russian).Search in Google Scholar
[2] Bodnarchuk V. G., Kaluzhnin L. A., Kotov V. N., Romov B. A., “Galois theory for Post algebras. II”, Kibernetika, 1969, №5, 1–9 (in Russian).10.1007/BF01267873Search in Google Scholar
[3] Yanov Yu. I., Muchnik A. A., “On the existence of k-valued closed classes without finite basis”, Doklady Akad. Nauk SSSR, 127, 1959, 44–46 (in Russian).Search in Google Scholar
[4] Kearnes K. A., KissE. W., Szendrei Á., “Growth rates of algebras, I: Pointed cube terms”, J. Australian Math. Soc., 101:1 (2016), 56–94.10.1017/S144678871500052XSearch in Google Scholar
[5] Kearnes K. A., Kiss E. W., Szendrei Á., “Growth rates of algebras, II: Wiegold dichotomy”, Int. J. Algebra and Computation, 25:4 (2015), 555–566.10.1142/S0218196715500101Search in Google Scholar
[6] Kearnes K. A., KissE. W., Szendrei Á., “Growth rates of algebras, III: finite solvable algebras”, Algebra universalis, 76:2(2016), 199–222.10.1007/s00012-016-0396-xSearch in Google Scholar
[7] Komkov S. A., “A new formulation of a criterion for the minimal logarithmic growth rate”, Moscow Univ. Math. Bulletin, 75:5 (2020), 219–220.10.3103/S0027132220050046Search in Google Scholar
[8] Komkov S. A., “Cardinality of generating sets for operations from the Post lattice classes”, Discrete Math. Appl., 29:3 (2019), 159–173.10.1515/dma-2019-0015Search in Google Scholar
[9] Komkov S. A., “On classes of functions of many-valued logic with minimal logarithmic growth rate”, Discrete Math. Appl., 30:4 (2020), 265–272.10.1515/dma-2020-0023Search in Google Scholar
[10] Lau D., Function Algebras on Finite Sets, Springer Science & Business Media, Berlin, Heidelberg, 2006, 668 pp.Search in Google Scholar
[11] Rosenberg I., “Über die funktionale Vollständigkeit in den mehrwertigen Logiken”, Pozpravy Českoslov. Akad. Ved., 80:3 (1970), 3–93.Search in Google Scholar
[12] Quick M., Ruškuc N., “Growth of generating sets for direct powers of classical algebraic structures”, J. Australian Math. Soc., 89:1 (2010), 105–126.10.1017/S1446788710001473Search in Google Scholar
[13] Wiegold J., “Growth sequences of finite groups”, J. Australian Math. Soc., 17:2 (1974), 133–141.10.1017/S1446788700016712Search in Google Scholar
[14] Zhuk D., “The lattice of all clones of self-dual functions in three-valued logic”, J. Multiple Valued Log. Soft Comput., 24:1–4 (2015), 251–316.Search in Google Scholar
[15] Zhuk D. N., “The predicate method to construct the Post lattice”, Discrete Math. Appl., 21:3 (2011), 329–344.10.1515/dma.2011.022Search in Google Scholar
[16] Zhuk D., “The size of generating sets of powers”, J. Comb. Theory, Ser. A, 167 (2019), 91–103.10.1016/j.jcta.2019.04.003Search 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