Abstract:Let $G = G(\cdot )$ be a commutative groupoid such that $\{(a,b,c) \in G^3$; $a\cdot bc \not =ab\cdot c\} = \{(a,b,c) \in G^3$; $a=b\not =c$ or $ a \not =b =c \}$. Then $G$ is determined uniquely up to isomorphism and if it is finite, then $ card (G) = 2^i$ for an integer $i\ge 0$.
Keywords: commutative groupoid, associative triples
AMS Subject Classification: 20N02, 05E99