Abstract
Based on the notion of an ideal of an iterative algebra, developed earlier, we introduce the concept of a simple iterative algebra of functions of a k-valued logic, that is, an algebra freed of proper ideals. It is found out that every simple algebra is contained in a maximal algebra and is naturally associated with some permutation group on a finite set. We specify a number of conditions on a permutation group under which a corresponding algebra is said to be maximal and simple. This allows us to completely list the maximal simple algebras for the case where k<-4.
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References
M. I. Mal'tsev,Iterative Post Algebras [in Russian], Nauka, Novosibirks (1976).
K. L. Safin, “Ideals of iterative algebras,”Sib. Mat. Zh., 6, No. 6, 1384–1391 (1995).
S. V. Yablonskii,Introduction to Discrete Mathematics [in Russian], Nauka, Moscow (1986).
P. P. Palfy and A. Szendrei, “Unary polynomials in algebras. II,”Contributions to General Algebra 2, Proc. Conf. Klagenfurt, Verlag Hölder-Pichler-Tempsky, Wien (1982), pp. 273–290.
R. Pöschel and L. A. Kalužnin,Funktionen und Relationenalgebren, VEB DVW, Berlin (1979).
Th. Ihringer, “A remark on clones and permutation groups,”Contributions to General Algebra 7, Proc. Conf. Vienna, Verlag Hölder-Pichler-Tempsky, Wien (1990), pp. 197–200.
M. Zichwolff, “Darstellung symmetrischer Strukturen durch Transversale,”Contributions to General Algebra 7, Proc. Conf. Vienna, Verlag Hölder-Pichler-Tempsky, Wien (1990), pp. 391–403.
Additional information
Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 460–477, July–August, 1998.
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Safin, K.L. Simple iterative algebras. Algebr Logic 37, 260–269 (1998). https://doi.org/10.1007/BF02671629
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DOI: https://doi.org/10.1007/BF02671629