Completely dissociative groupoids

Milton Braitt; David Hobby; Donald Silberger

Mathematica Bohemica (2012)

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

Abstract

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In a groupoid, consider arbitrarily parenthesized expressions on the k variables x 0 , x 1 , ⋯ 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 σ ( k ) . If u , v ∈ F σ ( k ) are distinct, the statement that u and v are equal for all values of x 0 , x 1 , ⋯ 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.

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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  1. Birkhoff, G., 10.1017/S0305004100013463, Proc. Camb. Philos. Soc. 31 (1935), 433-454. (1935) Zbl0013.00105DOI10.1017/S0305004100013463
  2. Braitt, M. S., Hobby, D., Silberger, D., Antiassociative groupoids, Preprint available. 
  3. Braitt, M. S., Silberger, D., Subassociative groupoids, Quasigroups Relat. Syst. 14 (2006), 11-26. (2006) Zbl1123.20059MR2268823
  4. Braitt, M. S., Hobby, D., Silberger, D., The sizings and subassociativity type of a groupoid, In preparation. 
  5. Burris, S., Sankappanavar, H. P., A Course in Universal Algebra, Springer (1981). (1981) Zbl0478.08001MR0648287
  6. Gould, H. W., Research Bibliography of Two Special Sequences, Combinatorial Research Institute, West Virginia University, Morgantown (1977). (1977) 
  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 ( 2 n - 2 ) ! / n ! ( n - 1 ) ! , Pr. Mat. 13 (1969), 91-96. (1969) Zbl0251.05005MR0249310

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