The Słupecki criterion by duality

Eszter K. Horváth

Discussiones Mathematicae - General Algebra and Applications (2001)

  • Volume: 21, Issue: 1, page 5-11
  • ISSN: 1509-9415

Abstract

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

How to cite

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Eszter K. Horváth. "The Słupecki criterion by duality." Discussiones Mathematicae - General Algebra and Applications 21.1 (2001): 5-11. <http://eudml.org/doc/287729>.

@article{EszterK2001,
abstract = {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 $A^\{k\}$ only. We show how the method applies for proving Słupecki’s classical theorem by generating diagonal relations from each pair of k-tuples.},
author = {Eszter K. Horváth},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {primal algebra; diagonal relation; Galois connection; Słupecki Criterion; Słupecki criterion; primality; functional completeness; operation-relation duality; diagonal relations},
language = {eng},
number = {1},
pages = {5-11},
title = {The Słupecki criterion by duality},
url = {http://eudml.org/doc/287729},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Eszter K. Horváth
TI - The Słupecki criterion by duality
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2001
VL - 21
IS - 1
SP - 5
EP - 11
AB - 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 $A^{k}$ only. We show how the method applies for proving Słupecki’s classical theorem by generating diagonal relations from each pair of k-tuples.
LA - eng
KW - primal algebra; diagonal relation; Galois connection; Słupecki Criterion; Słupecki criterion; primality; functional completeness; operation-relation duality; diagonal relations
UR - http://eudml.org/doc/287729
ER -

References

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  1. [1] K.A. Baker and A.F. Pixley, Polynomial Interpolation and the Chinese Remainder Theorem for Algebraic Systems, Math. Z. 143 (1975), 165-174. Zbl0292.08004
  2. [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. 
  3. [3] B. Csákány, Homogeneous algebras are functionally complete, AlgebraUniversalis 11 (1980), 149-158. Zbl0451.08004
  4. [4] A.L. Foster, An existence theorem for functionally complete universalalgebras, Math. Z. 71 (1959), 69-82. Zbl0084.26001
  5. [5] E. Fried and A.F. Pixley, The dual discriminator function in universalalgebra, Acta Sci. Math. (Szeged) 41 (1979), 83-100. Zbl0395.08001
  6. [6] D. Geiger, Closed systems of functions and predicates, Pacific J. Math. 27 (1968), 95-100. Zbl0186.02502
  7. [7] Th. Ihringer, Allgemeine Algebra, Teubner-Verlag, Stuttgart 1993. 
  8. [8] S.W. Jablonski and O.B. Lupanow, (eds.), Diskrete Mathematik und mathematische Fragen der Kybernetik, Akademie-Verlag, Berlin 1980. 
  9. [9] P.H. Krauss, On primal algebras, Algebra Universalis 2 (1972), 62-67. Zbl0265.08001
  10. [10] P.H. Krauss, On quasi primal algebras, Math. Z. 134 (1973), 85-89. Zbl0257.08004
  11. [11] R. Pöschel and L.A. Kaluznin, Funktionen- und Relationenalgebren, Deutschen Verlag der Wissenschaften, Berlin 1979. 
  12. [12] W. Sierpiński, Sur les fonctions de plusieurs variables, Fund. Math. 33 (1945), 169-173. Zbl0060.13111
  13. [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. [14] Á. Szendrei, Clones in Universal Algebra, Les Presses de l'Université de Montréal, Montreal 1986. 
  15. [15] H. Werner, Discriminator-Algebras, Akademie-Verlag, Berlin 1978. 
  16. [16] H. Werner, Eine Characterisierung funktional vollstandiger Algebren, Arch. Math. (Basel) 21 (1970), 381-385. Zbl0211.32003
  17. [17] S.V. Yablonski, Functional construction in the k-valued logic (Russian), Trudy Math. Inst. Steklov 51 (1958), 5-142. 

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