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«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». 2023. Vol 46

Criterion of Completeness and Submaximal Ultraclones for Linear Hyperfunctions of Rank 2

Author(s)
Ivan K. Sharankhaev1

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

Abstract
In recent years, the direction associated with the study of maps from a finite set A to the set of all subsets of the set A, including the empty one, has been intensively developing. Such mappings are called multifunctions on A, as well as hyperfunctions on A, if an empty subset is excluded from the subsets under consideration. It is not difficult to see that the so-called undefined or undefined functions, which are studied in many works, are most directly related to this field of research. The power of the set A is called the rank of multifunction or hyperfunction. Obviously, multifunctions and hyperfunctions generalize well-known functions of k-valued logic, however, it should be noted that the usual superposition of functions of k-valued logic is not suitable for multifunctions and hyperfunctions. Two types of superpositions are most often considered here, one of them leads to sets closed relative to the superposition, which are called multiclones and hyperclones, and for the second type of superposition, closed sets are called ultraclones and partial ultraclones. In this article, the elements of the rank 2 ultraclone lattice are considered. By now, all the maximum and minimum elements of this lattice are known. For example, Panteleev V.I. described all maximal ultraclones in the predicate language, which allowed us to prove the completeness criterion of an arbitrary system of hyperfunctions of rank 2. We managed to prove the completeness criterion in the maximal ultraclone of linear hyperfunctions of rank 2. Thus, all submaximal ultraclones of linear hyperfunctions are described.
About the Authors
Ivan K. Sharankhaev, Cand. Sci. (Phys.–Math.), Assoc. Prof., Dorzhi Banzarov Buryat State University, Ulan-Ude, 670000, Russian Federation, goran5@mail.ru
For citation
Sharankhaev I. K. Criterion of Completeness and Submaximal Ultraclones for Linear Hyperfunctions of Rank 2. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 46, pp. 121–129. (in Russian) https://doi.org/10.26516/1997-7670.2023.46.121
Keywords
hyperfunction, linear function, closed set, ultraclone, lattice
UDC
519.716
MSC
08A99
DOI
https://doi.org/10.26516/1997-7670.2023.46.121
References
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  2. Badmaev S.A., Dugarov A.E., Fomina I.V., Sharankhaev I.K. On some intervals in the lattice of ultraclones of rank 2. Siberian Electronic Mathematical Reports, 2021, vol. 18, no. 2, pp. 1210–1218. https://doi.org/10.33048/semi.2021.18.092
  3. Badmaev S.A., Dugarov A.E., Fomina I.V., Sharankhaev I.K. On two intervals in the lattice of partial ultraclones of rank 2. Siberian Electronic Mathematical Reports, 2023, vol. 20, no. 1, pp. 262–274.https://doi.org/10.33048/semi.2023.20.021
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