ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

Discussiones Mathematicae - General Algebra and Applications

Cite Score: 0.4

SJR: 0.203

SNIP: 0.562

MCQ: 0.12

Index Copernicus: 120.64

Discussiones Mathematicae - General Algebra and Applications


Discussiones Mathematicae General Algebra and Applications 21(1) (2001) 5-11


Eszter K. Horváth

Bolyai Institute, University of Szeged
Aradi vértanúk tere 1, H-6720 Szeged, Hungary



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 Ak only. We show how the method applies for proving Słupecki's classical theorem by generating diagonal relations from each pair of k-tuples.

Keywords: primal algebra, diagonal relation, Galois connection, Słupecki Criterion.

2000 Mathematics Subject Classification: 08A02, 08A40, 08A62, 06A15.


K.A. Baker and A.F. Pixley, Polynomial Interpolation and the Chinese Remainder Theorem for Algebraic Systems, Math. Z. 143 (1975), 165-174.
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. [3]
B. Csákány, Homogeneous algebras are functionally complete, AlgebraUniversalis 11 (1980), 149-158. [4]
A.L. Foster, An existence theorem for functionally complete universalalgebras, Math. Z. 71 (1959), 69-82. [5]
E. Fried and A.F. Pixley, The dual discriminator function in universalalgebra, Acta Sci. Math. (Szeged) 41 (1979), 83-100. [6]
D. Geiger, Closed systems of functions and predicates, Pacific J. Math. 27 (1968), 95-100. [7]
Th. Ihringer, Allgemeine Algebra, Teubner-Verlag, Stuttgart 1993. [8]
S.W. Jablonski and O.B. Lupanow, (eds.) Diskrete Mathematik und mathematische Fragen der Kybernetik, Akademie-Verlag, Berlin 1980. [9]
P.H. Krauss, On primal algebras, Algebra Universalis 2 (1972), 62-67. [10]
P.H. Krauss, On quasi primal algebras, Math. Z. 134 (1973), 85-89. [11]
R. Pöschel and L.A. Kaluznin, Funktionen- und Relationenalgebren, Deutschen Verlag der Wissenschaften, Berlin 1979 [12]
W. Sierpiński, Sur les fonctions de plusieurs variables, Fund. Math. 33 (1945), 169-173. [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). [14]
Á. Szendrei, Clones in Universal Algebra, Les Presses de l'Université de Montréal, Montreal 1986. [15]
H. Werner, Discriminator-Algebras, Akademie-Verlag, Berlin 1978. [16]
H. Werner, Eine Characterisierung funktional vollstandiger Algebren, Arch. Math. (Basel) 21 (1970), 381-385. [17]
S.V. Yablonski, Functional construction in the k-valued logic (Russian), Trudy Math. Inst. Steklov 51 (1958), 5-142.

Received 9 February 1998
Revised 6 November 2000
Revised 5 March 2001