рус eng
«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2025. Vol 51

Implicatively Precomplete Sets of Multioperations Defined on a Set of Three Elements

Author(s)
Vladimir I. Panteleev1,2

1Banzarov Buryat State University, Ulan-Ude, Russian Federation

2Irkutsk State University, Irkutsk, Russian Federation

Abstract
We study the completeness criterion on the set of rank-3 multioperations with respect to the implicative closure operator. The problem is a special case of the problem of finite classification of multioperations defined on an arbitrary set. A description of all precomplete sets is obtained. The expressive possibilities of the operator are described, including the conditions under which a set of operations implicatively generates all sets of multoperations. The obtained result can be used in the study of multioperations defined on an arbitrary set.
About the Authors
Vladimir I. Panteleev, Dr. Sci. (Phys.-Math.), Assoc. Prof., Banzarov Buryat State University, Ulan-Ude, 670000, Russian Federation; Irkutsk State University, Irkutsk, 664003, Russian Federation, vl.panteleyev@gmail.com
For citation

Panteleev V. I. Implicatively Precomplete Sets of Multioperations Defined on a Set of Three Elements. The Bulletin of Irkutsk State University. Series Mathematics, 2025, vol. 51, pp. 130–140. (in Russian)

https://doi.org/10.26516/1997-7670.2025.51.130

Keywords
closure, multioperation, closed set, composition, completeness, representability
UDC
519.716
MSC
03B50, 08A99
DOI
https://doi.org/10.26516/1997-7670.2025.51.130
References
  1. Danilchenko A.F. Parametric expressibility of functions of threevalued logic. Algebra and Logic, 1977, vol. 16, iss. 4, pp. 266–280. https://doi.org/10.1007/BF01669278 (in Russian)
  2. Kuznecov A.V. Means for detection of nondeducibility and inexpressibility. Logical Inference, Moscow, Nauka Publ., 1979, pp. 5–33.
  3. Marchenkov S.S. On the expressibility of functions of many-valued logic in some logical-functional classes. Discrete Math. Appl., 1999, vol. 9, no. 6, pp. 110–126. https://doi.org/10.1515/dma.1999.9.6.563 (in Russian)
  4. Marchenkov S.S. On the action of the implicative closure operator on the set of partial functions of the multivalued logic. Discrete Math. Appl., 2021, vol. 31, no. 3, pp. 155–164 https://doi.org/10.1515/dma-2021-0014 (in Russian)
  5. Marchenkov S. S. Logical extensions of the parametric closure operator. Discrete Math. Appl., 2023, vol. 33, no. 6, pp. 371–379. https://doi.org/10.1515/dma-2023-0033 (in Russian)
  6. Panteleev V.I., Taglasov E.S. ESI -closure of rank 2 multifunctions: completeness criterion, classification and types of bases. Intelligent systems. Theory and applications, 2021, vol. 25, no. 2, pp. 55–80. (in Russian)
  7. Riabets L.V. Parametric and positive closure operators on the set of rank 2 hyperfunctions. Intelligent systems. Theory and applications, 2016, vol. 20, no. 3, pp. 79–84.
  8. Riabets L.V. Parametric Closed Classes of Hyperfunctions on Two-Element Set. The Bulletin of Irkutsk State University. Series Mathematics, 2016, vol. 17, pp. 46–61. (in Russian)
  9. Sharankhaev I.K. On positive completeness and positively closed sets of multifunctions of rank 2. Siberian Electronic Mathematical Reports, 2023, vol. 20, no. 2. pp. 1313–1319. https://doi.org/10.33048/semi.2023.20.079 (in Russian)
  10. Barris S., Willard R. Finitely many primitive positive clones. Proc. Amer. Math. Soc., 1987, vol. 101, no. 3, pp. 427–430. https://doi.org/10.1090/S0002-9939-1987-0908642-5

Full text (russian)