A method is presented for proving primality and functional completeness theorems, which makes use of the operation-relation duality. By the result of Sierpiński, we have to investigate relations generated by the two-element subsets of only. We show how the method applies for proving Słupecki’s classical theorem by generating diagonal relations from each pair of k-tuples.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1022,
author = {Eszter K. Horv\'ath},
title = {The S\l upecki criterion by duality},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {21},
year = {2001},
pages = {5-11},
zbl = {0996.08001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1022}
}
Eszter K. Horváth. The Słupecki criterion by duality. Discussiones Mathematicae - General Algebra and Applications, Tome 21 (2001) pp. 5-11. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1022/
[000] [1] K.A. Baker and A.F. Pixley, Polynomial Interpolation and the Chinese Remainder Theorem for Algebraic Systems, Math. Z. 143 (1975), 165-174. | Zbl 0292.08004
[001] [2] V.G. Bodnarcuk, L.A. Kaluznin, V.N. Kotov, and B.A. Romov, Galois theory for Post algebras, I and II (Russian), Kibernetika (Kiev) 5 (1969), no. 3 p. 1-10 and no. 5, p. 1-9.
[002] [3] B. Csákány, Homogeneous algebras are functionally complete, AlgebraUniversalis 11 (1980), 149-158. | Zbl 0451.08004
[003] [4] A.L. Foster, An existence theorem for functionally complete universalalgebras, Math. Z. 71 (1959), 69-82. | Zbl 0084.26001
[004] [5] E. Fried and A.F. Pixley, The dual discriminator function in universalalgebra, Acta Sci. Math. (Szeged) 41 (1979), 83-100. | Zbl 0395.08001
[005] [6] D. Geiger, Closed systems of functions and predicates, Pacific J. Math. 27 (1968), 95-100. | Zbl 0186.02502
[006] [7] Th. Ihringer, Allgemeine Algebra, Teubner-Verlag, Stuttgart 1993.
[007] [8] S.W. Jablonski and O.B. Lupanow, (eds.), Diskrete Mathematik und mathematische Fragen der Kybernetik, Akademie-Verlag, Berlin 1980.
[008] [9] P.H. Krauss, On primal algebras, Algebra Universalis 2 (1972), 62-67. | Zbl 0265.08001
[009] [10] P.H. Krauss, On quasi primal algebras, Math. Z. 134 (1973), 85-89. | Zbl 0257.08004
[010] [11] R. Pöschel and L.A. Kaluznin, Funktionen- und Relationenalgebren, Deutschen Verlag der Wissenschaften, Berlin 1979.
[011] [12] W. Sierpiński, Sur les fonctions de plusieurs variables, Fund. Math. 33 (1945), 169-173. | Zbl 0060.13111
[012] [13] J. Słupecki, Completeness criterion for systems of many-valued propositional calculus (in Polish), C.R. des Séances de la Societé des Sciences et des Lettres de Varsovie Cl. II 32 (1939), 102-109., (English transl.: Studia Logica 30 (972), 153-157.
[013] [14] Á. Szendrei, Clones in Universal Algebra, Les Presses de l'Université de Montréal, Montreal 1986.
[014] [15] H. Werner, Discriminator-Algebras, Akademie-Verlag, Berlin 1978.
[015] [16] H. Werner, Eine Characterisierung funktional vollstandiger Algebren, Arch. Math. (Basel) 21 (1970), 381-385. | Zbl 0211.32003
[016] [17] S.V. Yablonski, Functional construction in the k-valued logic (Russian), Trudy Math. Inst. Steklov 51 (1958), 5-142.