Geodesic
Antiassociative groupoids
Mathematica Bohemica, Tome 142 (2017) no. 1, pp. 27-46
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Given a groupoid $\langle G, \star \rangle $, and $k \geq 3$, we say that $G$ is antiassociative if an only if for all $x_1, x_2, x_3 \in G$, $(x_1 \star x_2) \star x_3$ and $x_1 \star (x_2 \star x_3)$ are never equal. Generalizing this, $\langle G, \star \rangle $ is $k$-antiassociative if and only if for all $x_1, x_2, \ldots , x_k \in G$, any two distinct expressions made by putting parentheses in $x_1 \star x_2 \star x_3 \star \cdots \star x_k$ are never equal. \endgraf We prove that for every $k \geq 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.
Given a groupoid $\langle G, \star \rangle $, and $k \geq 3$, we say that $G$ is antiassociative if an only if for all $x_1, x_2, x_3 \in G$, $(x_1 \star x_2) \star x_3$ and $x_1 \star (x_2 \star x_3)$ are never equal. Generalizing this, $\langle G, \star \rangle $ is $k$-antiassociative if and only if for all $x_1, x_2, \ldots , x_k \in G$, any two distinct expressions made by putting parentheses in $x_1 \star x_2 \star x_3 \star \cdots \star x_k$ are never equal. \endgraf We prove that for every $k \geq 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.
DOI : 10.21136/MB.2017.0006-15
Classification : 08A99, 20N02, 68Q99, 68R15, 68T15
Keywords: groupoid; unification 
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Braitt, Milton; Hobby, David; Silberger, Donald. Antiassociative groupoids. Mathematica Bohemica, Tome 142 (2017) no. 1, pp. 27-46. doi: 10.21136/MB.2017.0006-15

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