Antiassociative Groupoids

TL;DR

This paper introduces the concept of antiassociative groupoids, proving their existence for any number of elements, and explores conditions under which different groupoid expressions are never equal.

Contribution

It establishes the existence of finite $k$-antiassociative groupoids for all $k \, \geq \, 3$ and generalizes the concept to other non-equal term pairs.

Findings

Existence of finite $k$-antiassociative groupoids for all $k \geq 3

Construction methods for such groupoids

Conditions for non-equality of different groupoid terms

Abstract

Given a groupoid $< G, \star >$, and $k \geq 3$, 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 $k \geq 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.