**
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

**Full text** available as PDF.

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade.
To activate your access, please contact Myris Trade at myris@myris.cz.