# Completely dissociative groupoids

Milton Braitt; David Hobby; Donald Silberger

Mathematica Bohemica (2012)

- Volume: 137, Issue: 1, page 79-97
- ISSN: 0862-7959

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topBraitt, Milton, Hobby, David, and Silberger, Donald. "Completely dissociative groupoids." Mathematica Bohemica 137.1 (2012): 79-97. <http://eudml.org/doc/247219>.

@article{Braitt2012,

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 $\lbrace 0,1 \rbrace $ where the groupoid operation is implication and NAND, respectively.},

author = {Braitt, Milton, Hobby, David, Silberger, Donald},

journal = {Mathematica Bohemica},

keywords = {groupoid; dissociative groupoid; generalized associative groupoid; formal product; reverse Polish notation (rPn); dissociative groupoids; generalized associative groupoids; formal products; reverse Polish notation},

language = {eng},

number = {1},

pages = {79-97},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Completely dissociative groupoids},

url = {http://eudml.org/doc/247219},

volume = {137},

year = {2012},

}

TY - JOUR

AU - Braitt, Milton

AU - Hobby, David

AU - Silberger, Donald

TI - Completely dissociative groupoids

JO - Mathematica Bohemica

PY - 2012

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 137

IS - 1

SP - 79

EP - 97

AB - 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 $\lbrace 0,1 \rbrace $ where the groupoid operation is implication and NAND, respectively.

LA - eng

KW - groupoid; dissociative groupoid; generalized associative groupoid; formal product; reverse Polish notation (rPn); dissociative groupoids; generalized associative groupoids; formal products; reverse Polish notation

UR - http://eudml.org/doc/247219

ER -

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