Abstract
Basic positively closed classes are intersections of positively precomplete classes. We prove that three-valued logic contains exactly 79 basic positively closed classes. Each class is described in terms of endomorphism semigroups.
Note
Originally published in Diskretnaya Matematika (2017) 29, №2, 40–52 (in Russian).
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: 16-01-00593
Funding statement: Research was partially supported by the Russian Basic Research Foundation, project 16-01-00593.
References
[1] Daniľchenko A. F., “Parametric expressibility of functions of three-valued logic”, 16:4 (1977), 266–280.10.1007/BF01669278Search in Google Scholar
[2] Daniľchenko A.F., “Parametrically closed classes of three-valued logic functions”, Izvestiya AN MSSR, 2 (1978), 13–20 (in Russian).Search in Google Scholar
[3] Kuznetsov A.V., “On means for detecting non-derivability and ineffectiveness”, Logical inference, Nauka, Moscow, 1979, 5–33 (in Russian).Search in Google Scholar
[4] Marchenkov S. S., “On expressibility of functions of many-valued logic in some logical-functional languages”, DiscreteMath. Appl., 9:6 (1999), 563–581.10.1515/dma.1999.9.6.563Search in Google Scholar
[5] Marchenkov S. S., “Criterion of positive completeness in three-valued logic”, J. Appl. Indust. Math. Ser. 1, 13:3 (2006), 27–39 (in Russian).Search in Google Scholar
[6] Marchenkov S. S., “Definition of positively closed classes by endomorphism semigroups”, DiscreteMath. Appl., 22:5-6 (2012), 511–520.10.1515/dma-2012-035Search in Google Scholar
[7] Nagorny A. S., “On the distribution of three-valued functions over pre-complete classes”,Moscow Univ. Comput.Math. Cyber., 36:3 (2012), 155–163.10.3103/S0278641912030053Search in Google Scholar
[8] Nagornyy A.S., “On the properties of precomplete classes in .P3”, Izv. VUZ’ov. Povolzhskiy region. Fiz.-mat. nauki, 2012, №2, 16–24 (in Russian).Search in Google Scholar
[9] Yanov Yu.I., Muchnik A.A., “On the existence of k -valued closed classes without basis”, Doklady AN SSSR, 127:1 (1959), 44–46 (in Russian).Search in Google Scholar
[10] Daniľčenko A.F., “On parametrical expressibility of the functions of k-valued logic”, Colloq. Math. Soc. J.Bolyai, 28 (1981), 147–159.Search in Google Scholar
[11] Post E.L., “Introduction to a general theory of elementary propositions”, Amer. J. Math., 43:4 (1921), 163–185.10.2307/2370324Search in Google Scholar
[12] Post E.L., “Two-valued iterative systems of mathematical logic”, Annals of Math. Studies, 5 (1941), 1–122.10.1515/9781400882366Search in Google Scholar
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