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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Finitely many primitive positive clones
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by S. Burris and R. Willard PDF
Proc. Amer. Math. Soc. 101 (1987), 427-430 Request permission

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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • 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