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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
DOI: https://doi.org/10.1090/S0002-9939-1987-0908642-5
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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DOI: https://doi.org/10.1090/S0002-9939-1987-0908642-5
Keywords: Primitive positive, clone, model companion, universal Horn, congruence
Article copyright: © Copyright 1987 American Mathematical Society

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