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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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