Affine spaces over GF(3) (Q1868249)

For an algebra \textbf{A}, \(p_n({\mathbf A})\) denotes the number of essentially \(n\)-ary term functions over \textbf{A}. It was shown by \textit{G. Grätzer} and \textit{R. Padmanabhan} [Proc. Am. Math. Soc. 28, 75-80 (1971; Zbl 0215.34501)] that an algebra \textbf{A} is a nontrivial affine space over GF(3) if and only if \(p_n({\mathbf A})=(2^n-(-1)^n)/3\) for \(n=1,2,\dots\). The author of this paper gives various results where extra conditions on \textbf{A} enable a weakening of the conditions on \(p_n({\mathbf A})\). The particular result given in this note is that an idempotent algebra \((G,s)\) of type (3) is a nontrivial affine space over GF(3) if and only if \(p_4(G,s)=5\).