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 $\mathbf{u},\mathbf{v}\subseteq F^{\sigma }(k)$, the claim, that $\mathbf{u}$ and $\mathbf{v}$ 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.

arXiv comment29 pages