On idempotent, commutative, and nonassociative groupoids

Authors:
G. GrĂ¤tzer and R. Padmanabhan

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

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Abstract | References | Similar Articles | Additional Information

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.

- J. Dudek,
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*On the number of polynomials of a universal algebra. I*, Colloq. Math.**22**(1970), 9â€“11. MR**294216**, DOI https://doi.org/10.4064/cm-22-1-9-11 - J. PĹ‚onka,
*On the number of independent elements in finite abstract algebras having a binary operation*, Colloq. Math.**14**(1966), 189â€“201. MR**191861**, DOI https://doi.org/10.4064/cm-14-1-189-201 - J. PĹ‚onka,
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*Universal algebra*, Trends in Lattice Theory, Van Nostrand, Princeton, N. J., 1969.

*On algebras with $n$ distinct essentially $n$-ary operations*, Algebra Universalis (to appear). ---,

*On algebras with at most $n$ distinct essentially $n$-ary operations*, Algebra Universalis (to appear).

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Additional Information

Keywords:
<IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img5.gif" ALT="$n$">-ary polynomials and essentially <IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img6.gif" ALT="$n$">-ary polynomials of a universal algebra,
representable sequence,
minimal extension property,
idempotent reduct of an abelian group

Article copyright:
© Copyright 1971
American Mathematical Society