Antiassociative Groupoids

Abstract

Given a groupoid $<G,\star >$, and $k\ge 3$, we say that $G$ is antiassociative iff 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, $<G,\star >$ is $k$-antiassociative iff for all $x_1,x_2,...x_k\in G$, any two distinct expressions made by putting parentheses in $x_1\star x_2\star x_3\star ...x_k$ are never equal. We prove that for every $k\ge 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.

20 pages, 2 figures. Submitted to Journal of Algebra