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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
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