Galois theory for clones and superclones. (English. Russian original) Zbl 1346.08002
Discrete Math. Appl. 26, No. 4, 227-238 (2016); translation from Diskretn. Mat. 27, No. 4, 79-93 (2015).
Summary: We study clones (closed sets of operations that contain projections) and superclones on finite sets. According to A. I. Mal’tsev [Algebra Logika 5, No. 2, 5–24 (1966; Zbl 0275.08001)] 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.
MSC:
| 08A40 | Operations and polynomials in algebraic structures, primal algebras |
| 06A15 | Galois correspondences, closure operators (in relation to ordered sets) |
Citations:
Zbl 0275.08001References:
| [1] | Yablonskiy S. V. “On superpositions of functions of the algebra logic”, Matem. sb., 30 (1952), 329-348 (in Russian).; Yablonskiy, S. V., “On superpositions of functions of the algebra logic”, Matem. sb, 30, 329-348 (1952) · Zbl 0046.00601 |
| [2] | Yablonskiy S. V. “Functional constructions in a k-valued logic”, Trudy Mat. Inst. Steklov, 51 (1958), 5-142 (in Russian).; Yablonskiy, S. V., “Functional constructions in a k-valued logic”, Trudy Mat. Inst. Steklov, 51, 5-142 (1958) · Zbl 0092.25101 |
| [3] | Mal’tsev A. I. “Iterative algebras and Post varieties”, Algebra and Logic, 5:2 (1966), 3-9 (in Russian).; Mal’tsev, A. I., “Iterative algebras and Post varieties”, Algebra and Logic, 5, 2, 3-9 (1966) · Zbl 0248.08006 |
| [4] | Cohn P. M, Universal Algebra, Harper and Row, New York, 1965.; Cohn, P. M., Universal Algebra (1965) · Zbl 0141.01002 |
| [5] | Lau D., Function algebras on finite sets. A basic course on many-valued logic and clone theory, Springer-VerlagBerlin 2006, 668 pp.; Lau, D., A basic course on many-valued logic and clone theory, 668 (2006) · Zbl 1105.08001 |
| [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).; Bodnarchuk, V. G.; Kaluzhnin, L. A.; Kotov, V. N.; Romov, B. A., “Galois theory for Post algebras”, Kibernetika, 3, 1-10 (1969) · Zbl 0212.31802 |
| [7] | Poscher R., Kaluznin L. A., Funktionen- und Relationenalgebren, Ein Kapitel der diskreten Mathematik, VEB DVWBerlin 1979, 237 pp.; Poscher, R.; Kaluznin, L. A., Ein Kapitel der diskreten Mathematik, 237 (1979) · Zbl 0418.03044 |
| [8] | Peryazev N. A., “Underdetermined partial Boolean functions”, Mater. XV Mezhdunar. konf. ≪Problemy teoreticheskoy kibernetiki≫, 2008, 92 (in Russian).; Peryazev, N. A., “Underdetermined partial Boolean functions”, Mater. XV Mezhdunar. konf. ≪Problemy teoreticheskoy kibernetiki≫, 92 (2008) |
| [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).; Peryazev, N. A., “Clones, co-clones, hyperclones and superclones”, Uchenye zapiski Kazanskogo gos. un-ta. Seriya: Fiziko-matem. nauki, 151, 2, 120-125 (2009) · Zbl 1220.08003 |
| [10] | Peryazev N. A., “Multioperations and superclones”, Mezhdunar. konf. ≪Mal’tsevskie chteniya≫. Tezisy dokl., 2011, 84 (in Russian).; Peryazev, N. A., “Multioperations and superclones”, Mezhdunar. konf. ≪Mal’tsevskie chteniya≫. Tezisy dokl, 84 (2011) |
| [11] | Peryazev N. A., “Galois theory for clones and superclones”, Materialy XVI Mezhdunar. konf. ≪Problemy teoreticheskoy kibernetiki≫, 2011, 359-363 (in Russian).; Peryazev, N. A., “Galois theory for clones and superclones”, Materialy XVI Mezhdunar. konf. ≪Problemy teoreticheskoy kibernetiki≫, 359-363 (2011) |
| [12] | Ore O., Graph theory, AMS 1962, 270 pp.; Ore, O., Graph theory, 270 (1962) · Zbl 0105.35401 |
| [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).; Kuznetsov, A. V., In book:, Logicheskiy vyvod, 5-33 (1979) |
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