# zbMATH — the first resource for mathematics

Completely dissociative groupoids. (English) Zbl 1249.20075
For a fixed integer $$k\geq 3$$, consider a groupoid identity $$x_1\cdots x_k=x_1\cdots x_k$$ where the left hand side and the right hand side are parenthesized in different ways. A groupoid $$G$$ is $$k$$-dissociative if no such identity holds in $$G$$, and it is completely dissociative if it is $$k$$-dissociative for every $$k\geq 3$$.
The paper introduces some elementary techniques by which one can often decide whether a small groupoid is completely dissociative. For instance, it is shown that among the $$16$$ groupoids of order $$2$$ precisely $$6$$ are completely dissociative, including the groupoids expressing the truth table for implication and the logical operation NAND.
##### MSC:
 20N02 Sets with a single binary operation (groupoids) 08A02 Relational systems, laws of composition
Full Text: