Geodesic
On a class of commutative groupoids determined by their associativity triples
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 2, pp. 199-201.

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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 $i\ge 0$.
Classification : 05B15, 05E99, 20L05, 20N02
Keywords: commutative groupoid; associative triples 
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     author = {Dr\'apal, Ale\v{s}},
     title = {On a class of commutative groupoids determined by their associativity triples},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {199--201},
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Drápal, Aleš. On a class of commutative groupoids determined by their associativity triples. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 2, pp. 199-201. http://geodesic.mathdoc.fr/item/CMUC_1993__34_2_a0/