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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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On idempotent, commutative, and nonassociative groupoids
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by G. Grätzer and R. Padmanabhan PDF
Proc. Amer. Math. Soc. 28 (1971), 75-80 Request permission

Abstract:

For an algebra $\mathfrak {A} = \left \langle {A;F} \right \rangle$ and for $n \geqq 2$, let ${p_n}(\mathfrak {A})$ denote the number of essentially $n$-ary polynomials of $\mathfrak {A}$. J. Dudek has shown that if $\mathfrak {A}$ is an idempotent and nonassociative groupoid then ${p_n}(\mathfrak {A}) \geqq n$ for all $n > 2$. In this paper this result is improved for the commutative case to show that for such groupoids $\mathfrak {A},{p_n}(\mathfrak {A}) \geqq \frac {1}{3}({2^n} - {( - 1)^n})$ for all $n \geqq 2$ (Theorem 1) and that this is the best possible result. Those groupoids for which this lower bound is attained are completely characterized. In fact, the relevant result proved below is much stronger (Theorem 3). From these and other known results it is deduced that the sequence $\left \langle {0,0,1,3} \right \rangle$ has the minimal extension property.
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 28 (1971), 75-80
  • MSC: Primary 20.95
  • DOI: https://doi.org/10.1090/S0002-9939-1971-0276393-7
  • MathSciNet review: 0276393