# Completely dissociative groupoids

Mathematica Bohemica (2012)

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

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## Abstract

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In a groupoid, consider arbitrarily parenthesized expressions on the $k$ variables ${x}_{0},{x}_{1},\cdots {x}_{k-1}$ 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 ${F}^{\sigma }\left(k\right)$. If $u,v\in {F}^{\sigma }\left(k\right)$ are distinct, the statement that $u$ and $v$ are equal for all values of ${x}_{0},{x}_{1},\cdots {x}_{k-1}$ 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 $\left\{0,1\right\}$ where the groupoid operation is implication and NAND, respectively.

## How to cite

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Braitt, 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 -

## References

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2. Braitt, M. S., Hobby, D., Silberger, D., Antiassociative groupoids, Preprint available.
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4. Braitt, M. S., Hobby, D., Silberger, D., The sizings and subassociativity type of a groupoid, In preparation.
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7. Quine, W. V., 10.2307/2307285, Am. Math. Mon. 62 (1955), 627-631. (1955) Zbl0068.24209MR0075886DOI10.2307/2307285
8. Quine, W. V., Selected Logic Papers: Enlarged Edition, Harvard University Press (1995). (1995) MR1329994
9. Silberger, D. M., Occurrences of the integer $\left(2n-2\right)!/n!\left(n-1\right)!$, Pr. Mat. 13 (1969), 91-96. (1969) Zbl0251.05005MR0249310

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