Mathematica Bohemica, Vol. 142, No. 1, pp. 27-46, 2017
Antiassociative groupoids
Milton Braitt, David Hobby, Donald Silberger
Received February 4, 2015. First published October 31, 2016.
Abstract: Given a groupoid $\langle G, \star\rangle$, and $k \geq3$, 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 $x_1, x_2, \ldots, x_k \in G$, 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 $k \geq3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.
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Affiliations: Milton Braitt, Universidade Federal de Santa Catarina, R. Eng. Agronômico Andrei Cristian Ferreira, Trindade, Florianópolis, Santa Catarina 88040-900, Brazil, e-mail: m.braitt@ufsc.br; David Hobby, Donald Silberger, State University of New York, 1 Hawk Drive, New Paltz, NY 12561, USA, e-mail: hobbyd@newpaltz.edu, silbergd@newpaltz.edu
