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Antiassociative groupoids. (English) Zbl 1424.20085
Authors’ abstract: Given a groupoid \(\langle G,\star\rangle\), and \(k\geq 3\), we say that \(G\) is antiassociative if an only if for all \(x_1, x_2, x_3\in G\), \((x_1\star x_2)\star x_3\) and \(x_1\star (x_2\star x_3)\) are never equal. Generalizing this, \(\langle G,\star\rangle\) is \(k\)-antiassociative if and only if for all \(x_1, x_2,\dots ,x_k\in G\), any two distinct expressions made by putting parentheses in \(x_1\star x_2\star x_3\star\cdots\star x_k\) are never equal. We prove that for every \(k\geq 3\), there exist finite groupoids that are \(k\)-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.
MSC:
20N02 Sets with a single binary operation (groupoids)
Keywords:
goupoid; unification
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