Completely dissociative groupoids

TL;DR

This paper investigates the properties of groupoids where no generalized associative laws hold, providing examples and proving various results about their dissociative nature.

Contribution

It introduces the concept of completely dissociative groupoids and proves several results demonstrating their properties and examples.

Findings

Examples of completely dissociative groupoids on {0,1} with implication and NAND.

Many groupoids do not satisfy any generalized associative law.

Theoretical results characterizing dissociative groupoids.

Abstract

Consider arbitrarily parenthesized expressions on the $k$ variables $x_0, x_1, ..., x_{k-1}$, where each $x_i$ appears exactly once and in the order of their indices. We call these expressions {\em formal $k$--products}. $F^\sigma(k)$ denotes the set of formal $k$--products. For ${{\bf u},{\bf v}}\subseteq F^\sigma(k)$, the claim, that ${\bf u}$ and ${\bf v}$ produce equal elements in a groupoid $G$ for all values assumed in $G$ by the variables $x_i$, attributes to $G$ a {\em generalized associative law}. Many groupoids are {\em completely dissociative}; i.e., no generalized associative law holds for them; two examples are the groupoids on ${0,1}$ whose binary operations are implication and NAND. We prove a variety of results of that flavor.