Antiassociative groupoids
Milton Braitt, David Hobby, D. M. Silberger · Mathematica Bohemica · 2016
Given a groupoid $\langle G, \star\rangle$, and $k \geq3$, 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. We prove that for every $k \geq3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.