Galois theory for clones and superclones
-
Nikolay A. Peryazev
and
Ivan K. Sharankhaev
Abstract
We study clones (closed sets of operations that contain projections) and superclones on finite sets. According to A. I. Mal’tsev a clone may be considered as an algebra. If we replace algebra universe with a set of multioperations and add the operation of simplest equation solvability then we will obtain an algebra called a superclone. The paper establishes Galois connection between clones and superclones.
Originally published in Diskretnaya Matematika (2015) 27, №4, 79–93 (in Russian).
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: 12-01-000351a
Funding statement: Research was supported by RFBR, project number 12-01-000351a
References
[1] Yablonskiy S. V. “On superpositions of functions of the algebra logic”, Matem. sb., 30 (1952), 329–348 (in Russian).Search in Google Scholar
[2] Yablonskiy S. V. “Functional constructions in a k-valued logic”, Trudy Mat. Inst. Steklov, 51 (1958), 5–142 (in Russian).Search in Google Scholar
[3] Mal’tsev A. I. “Iterative algebras and Post varieties”, Algebra and Logic, 5:2 (1966), 3–9 (in Russian).Search in Google Scholar
[4] Cohn P. M, Universal Algebra, Harper and Row, New York, 1965.Search in Google Scholar
[5] Lau D., Function algebras on finite sets. A basic course on many-valued logic and clone theory, Springer-VerlagBerlin 2006, 668 pp.Search in Google Scholar
[6] Bodnarchuk V. G., Kaluzhnin L. A., Kotov V.N., Romov B. A., “Galois theory for Post algebras”, Kibernetika, 1969, No 3, p. 1–10; No 5, 1–9 (in Russian).10.1007/BF01070906Search in Google Scholar
[7] Poscher R., Kaluznin L. A., Funktionen- und Relationenalgebren, Ein Kapitel der diskreten Mathematik, VEB DVWBerlin 1979, 237 pp.10.1007/978-3-0348-5547-1_5Search in Google Scholar
[8] Peryazev N. A., “Underdetermined partial Boolean functions”, Mater. XV Mezhdunar. konf. ≪Problemy teoreticheskoy kibernetiki≫, 2008, 92 (in Russian).Search in Google Scholar
[9] Peryazev N. A., “Clones, co-clones, hyperclones and superclones”, Uchenye zapiski Kazanskogo gos. un-ta. Seriya: Fiziko-matem. nauki, 151:2 (2009), 120–125 (in Russian).Search in Google Scholar
[10] Peryazev N. A., “Multioperations and superclones”, Mezhdunar. konf. ≪Mal’tsevskie chteniya≫. Tezisy dokl., 2011, 84 (in Russian).Search in Google Scholar
[11] Peryazev N. A., “Galois theory for clones and superclones”, Materialy XVI Mezhdunar. konf. ≪Problemy teoreticheskoy kibernetiki≫, 2011, 359–363 (in Russian).10.1515/dma-2016-0020Search in Google Scholar
[12] Ore O., Graph theory, AMS 1962, 270 pp.Search in Google Scholar
[13] Kuznetsov A. V., “On the methods to detect the non-deducibility or non-expressibility”, In book:, Logicheskiy vyvod, NaukaMoscow 1979, 5–33 (in Russian).Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Research Article
- Characterization of almost perfect nonlinear functions in terms of subfunctions
- Research Article
- Estimates for distribution of the minimal distance of a random linear code
- Research Article
- The distributions of interrecord fillings
- Research Article
- Galois theory for clones and superclones
- Research Article
- Overgroups of order 2n additive regular groups of a residue ring and of a vector space
Articles in the same Issue
- Frontmatter
- Research Article
- Characterization of almost perfect nonlinear functions in terms of subfunctions
- Research Article
- Estimates for distribution of the minimal distance of a random linear code
- Research Article
- The distributions of interrecord fillings
- Research Article
- Galois theory for clones and superclones
- Research Article
- Overgroups of order 2n additive regular groups of a residue ring and of a vector space
