On idempotent, commutative, and nonassociative groupoids

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.