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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On idempotent, commutative, and nonassociative groupoids

Authors: G. Grätzer and R. Padmanabhan
Journal: Proc. Amer. Math. Soc. 28 (1971), 75-80
MSC: Primary 20.95
MathSciNet review: 0276393
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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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PII: S 0002-9939(1971)0276393-7
Keywords: $ n$-ary polynomials and essentially $ n$-ary polynomials of a universal algebra, representable sequence, minimal extension property, idempotent reduct of an abelian group
Article copyright: © Copyright 1971 American Mathematical Society