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Keywords:
groupoid; unification
groupoid; unification
Summary:
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.
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