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.

Milton Braitt, Universidade Federal de Santa Catarina, R. Eng. Agronômico Andrei Cristian Ferreira, Trindade, Florianópolis, Santa Catarina 88040-900, Brazil, e-mail:; David Hobby, Donald Silberger, State University of New York, 1 Hawk Drive, New Paltz, NY 12561, USA, e-mail:,

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.

Keywords: groupoid; unification

Classification (MSC 2010): 20N02, 08A99, 68Q99, 68T15

DOI: 10.21136/MB.2017.0006-15

Full text available as PDF.

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