MATHEMATICA BOHEMICA, Vol. 137, No. 1, pp. 79-97, 2012

# Completely dissociative groupoids

## Milton Braitt, David Hobby, Donald Silberger

Milton S. Braitt, Departamento de Matemática, Universidade Federal de Santa Catarina, Cidade Universitária, Florianopólis, SC 88040-900, Brasil, e-mail: MSBraitt@mtm.ufsc.br; David Hobby, Department of Mathematics, State University of New York at New Paltz, NY 12561, U.S.A., e-mail: hobbyd@newpaltz.edu; Donald Silberger, Department of Mathematics, State University of New York at New Paltz, NY 12561, U.S.A., e-mail: DonSilberger@hvc.rr.com

Abstract: In a groupoid, consider arbitrarily parenthesized expressions on the $k$ variables $x_0, x_1, \dots x_{k-1}$ where each $x_i$ appears once and all variables appear in order of their indices. We call these expressions $k$-ary formal products, and denote the set containing all of them by $F^\sigma(k)$. If $u,v \in F^\sigma(k)$ are distinct, the statement that $u$ and $v$ are equal for all values of $x_0, x_1, \dots x_{k-1}$ is a generalized associative law. Among other results, we show that many small groupoids are completely dissociative, meaning that no generalized associative law holds in them. These include the two groupoids on $\{ 0,1 \}$ where the groupoid operation is implication and NAND, respectively.

Keywords: groupoid, dissociative groupoid, generalized associative groupoid, formal product, reverse Polish notation (rPn)

Classification (MSC 2010): 20N02, 05A99, 08A99, 08B99, 08C10

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