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Keywords:
commutative groupoid; associative triples
commutative groupoid; associative triples
Summary:
Let $G = G(\cdot)$ be a commutative groupoid such that $\{(a,b,c) \in G^3$; $a\cdot bc \ne ab\cdot c\} = \{(a,b,c) \in G^3$; $a=b\ne c$ or $ a \ne b =c \}$. Then $G$ is determined uniquely up to isomorphism and if it is finite, then $\operatorname{card}(G) = 2^i$ for an integer $i\ge 0$.
References:
[1] Drápal A., Kepka T.: Sets of associative triples. Europ. J. Combinatorics 6 (1985), 227-231. MR 0818596
[2] Drápal A.: Groupoids with non-associative triples on the diagonal. Czech. Math. Journal 35 (1985), 555-564. MR 0809042
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