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Finitely many primitive positive clones

Authors: S. Burris and R. Willard
Journal: Proc. Amer. Math. Soc. 101 (1987), 427-430
MSC: Primary 08A40; Secondary 03C50
MathSciNet review: 908642
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Abstract: Given a finite set $ A$ there are only finitely many sequences of the form $ {\left\langle {\operatorname{Con}({{\mathbf{A}}^n})} \right\rangle _{n \geq 1}}$ or $ {\left\langle {\operatorname{Hom}({{\mathbf{A}}^n},{\mathbf{A}})} \right\rangle _{n \geq 1}}$, where $ {\mathbf{A}}$ is any algebra on $ A$. From this we derive the fact that there are only finitely many primitive positive clones on $ A$, which solves a problem posed by A. F. Danil'čenko in the 1970s. Consequently there are only finitely many model companions for universal Horn classes generated by an algebra of a given finite size.

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  • [1] M. Albert, A preservation theorem for ec-structures with applications, J. Symbolic Logic (to appear). MR 902990 (89a:03066)
  • [2] V. G. Bodnarčuk, L. A. Kalužnin, V. A. Kotov, and V. A. Romov, Galois theory for Post algebras. I, II, Kibernetika (Kiev) 5 (1969), no. 3, 1-10; no. 5, 1-9 (Russian); English transl., Cybernetics 5 (1969), 243-252, 531-539. MR 0300895 (46:55)
  • [3] S. Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Texts in Math., no. 78, Springer-Verlag, New York, 1981. MR 648287 (83k:08001)
  • [4] S. Burris and H. Werner, Sheaf constructions and their elementary properties, Trans. Amer. Math. Soc. 248 (1979), 269-309. MR 522263 (82d:03049)
  • [5] A. F. Danil'čenko, Parametric expressibility of functions of three-valued logic, Algebra i Logika 16 (1977), 397-416 (Russian); English transl., Algebra and Logic 16 (1977), 266-280. MR 516292 (80c:03024)
  • [6] -, On parametrical expressibility of the functions of $ k$-valued logic, Finite Algebra and Multiple-Valued Logic (B. Csákány and I. Rosenberg, eds.), Colloq. Math. Soc. János Bolyai, vol. 28, North-Holland, 1981, pp. 147-159.
  • [7] D. Geiger, Closed systems of functions and predicates, Pacific J. Math. 27 (1968), 95-100. MR 0234893 (38:3207)
  • [8] R. Pöschel, Die funktionale Vollständigkeit von Funktionenklassen über einer Familie endlicher Mengen, Z. Math. Logik Grundlagen Math. 20 (1974), 537-550. MR 0387000 (52:7849)
  • [9] J. Shoenfield, Mathematical logic, Addison-Wesley, Reading, Mass., 1967. MR 0225631 (37:1224)
  • [10] L. Szabó, Concrete representation of related structures of universal algebras. I, Acta Sci. Math. (Szeged) 40 (1978), 175-184. MR 0480264 (58:443)
  • [11] -, On the lattice of clones acting bicentrally, Acta Cybernet. 6 (1984), 381-388. MR 758493 (85j:08012)

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Keywords: Primitive positive, clone, model companion, universal Horn, congruence
Article copyright: © Copyright 1987 American Mathematical Society

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