Completely 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.