Affine spaces over GF(4) (Q1272211)

An algebra \(A\) is said to represent the sequence \((p_0,p_1,\dots)\) if the number of essentially \(n\)-ary term operations in \(A\) is equal to \(p_n\). It was shown by \textit{G. Grätzer} and \textit{R. Padmanabhan} [Proc. Am. Math. Soc. 28, 75-80 (1971; Zbl 0215.34501)] that \((G,\cdot)\) is a nontrivial affine space over \(GF(3)\) if and only if \((G,\cdot)\) represents the sequence \((0,1,1,3,5).\) The authors prove the following Theorem: A groupoid \((G,\cdot)\) is a nontrivial affine space over \(GF(4)\) if and only if \((G,\cdot)\) represents the sequence \((0,1,2,7)\).