| Title:
|
Completely dissociative groupoids (English) |
| Author:
|
Braitt, Milton |
| Author:
|
Hobby, David |
| Author:
|
Silberger, Donald |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
137 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
79-97 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. \endgraf 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. (English) |
| Keyword:
|
groupoid |
| Keyword:
|
dissociative groupoid |
| Keyword:
|
generalized associative groupoid |
| Keyword:
|
formal product |
| Keyword:
|
reverse Polish notation (rPn) |
| MSC:
|
05A99 |
| MSC:
|
08A99 |
| MSC:
|
08B99 |
| MSC:
|
08C10 |
| MSC:
|
20N02 |
| idZBL:
|
Zbl 1249.20075 |
| idMR:
|
MR2978447 |
| DOI:
|
10.21136/MB.2012.142789 |
| . |
| Date available:
|
2012-04-19T00:03:19Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142789 |
| . |
| Reference:
|
[1] Birkhoff, G.: On the structure of abstract algebras.Proc. Camb. Philos. Soc. 31 (1935), 433-454. Zbl 0013.00105, 10.1017/S0305004100013463 |
| Reference:
|
[2] Braitt, M. S., Hobby, D., Silberger, D.: Antiassociative groupoids.Preprint available. |
| Reference:
|
[3] Braitt, M. S., Silberger, D.: Subassociative groupoids.Quasigroups Relat. Syst. 14 (2006), 11-26. Zbl 1123.20059, MR 2268823 |
| Reference:
|
[4] Braitt, M. S., Hobby, D., Silberger, D.: The sizings and subassociativity type of a groupoid.In preparation. |
| Reference:
|
[5] Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra.Springer (1981). Zbl 0478.08001, MR 0648287 |
| Reference:
|
[6] Gould, H. W.: Research Bibliography of Two Special Sequences.Combinatorial Research Institute, West Virginia University, Morgantown (1977). |
| Reference:
|
[7] Quine, W. V.: A way to simplify truth functions.Am. Math. Mon. 62 (1955), 627-631. Zbl 0068.24209, MR 0075886, 10.2307/2307285 |
| Reference:
|
[8] Quine, W. V.: Selected Logic Papers: Enlarged Edition.Harvard University Press (1995). MR 1329994 |
| Reference:
|
[9] Silberger, D. M.: Occurrences of the integer $(2n-2)!/n!(n-1)!$.Pr. Mat. 13 (1969), 91-96. Zbl 0251.05005, MR 0249310 |
| . |
