Antiflexible Latin directed triple systems
Commentationes Mathematicae Universitatis Carolinae (2015)
- Volume: 56, Issue: 4, page 417-431
- ISSN: 0010-2628
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topKozlik, Andrew R.. "Antiflexible Latin directed triple systems." Commentationes Mathematicae Universitatis Carolinae 56.4 (2015): 417-431. <http://eudml.org/doc/276195>.
@article{Kozlik2015,
abstract = {It is well known that given a Steiner triple system one can define a quasigroup operation $\cdot $ upon its base set by assigning $x \cdot x = x$ for all $x$ and $x \cdot y = z$, where $z$ is the third point in the block containing the pair $\lbrace x,y\rbrace $. The same can be done for Mendelsohn triple systems, where $(x,y)$ is considered to be ordered. But this is not necessarily the case for directed triple systems. However there do exist directed triple systems, which induce a quasigroup under this operation and these are called Latin directed triple systems. The quasigroups associated with Steiner and Mendelsohn triple systems satisfy the flexible law $y \cdot (x \cdot y) = (y \cdot x) \cdot y$ but those associated with Latin directed triple systems need not. In this paper we study the Latin directed triple systems where the flexible identity holds for the least possible number of ordered pairs $(x, y)$. We describe their geometry, present a surprisingly simple cyclic construction and prove that they exist if and only if the order $n$ is $n\equiv 0$ or $1\hspace\{4.44443pt\}(\@mod \; 3)$ and $n\ge 13$.},
author = {Kozlik, Andrew R.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {directed triple system; quasigroup},
language = {eng},
number = {4},
pages = {417-431},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Antiflexible Latin directed triple systems},
url = {http://eudml.org/doc/276195},
volume = {56},
year = {2015},
}
TY - JOUR
AU - Kozlik, Andrew R.
TI - Antiflexible Latin directed triple systems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 4
SP - 417
EP - 431
AB - It is well known that given a Steiner triple system one can define a quasigroup operation $\cdot $ upon its base set by assigning $x \cdot x = x$ for all $x$ and $x \cdot y = z$, where $z$ is the third point in the block containing the pair $\lbrace x,y\rbrace $. The same can be done for Mendelsohn triple systems, where $(x,y)$ is considered to be ordered. But this is not necessarily the case for directed triple systems. However there do exist directed triple systems, which induce a quasigroup under this operation and these are called Latin directed triple systems. The quasigroups associated with Steiner and Mendelsohn triple systems satisfy the flexible law $y \cdot (x \cdot y) = (y \cdot x) \cdot y$ but those associated with Latin directed triple systems need not. In this paper we study the Latin directed triple systems where the flexible identity holds for the least possible number of ordered pairs $(x, y)$. We describe their geometry, present a surprisingly simple cyclic construction and prove that they exist if and only if the order $n$ is $n\equiv 0$ or $1\hspace{4.44443pt}(\@mod \; 3)$ and $n\ge 13$.
LA - eng
KW - directed triple system; quasigroup
UR - http://eudml.org/doc/276195
ER -
References
top- Brouwer A.E., Schrijver A., Hanani H., 10.1016/0012-365X(77)90037-1, Discrete Math. 20 (1977), 1–10. Zbl1093.05008MR0465894DOI10.1016/0012-365X(77)90037-1
- Colbourn C.J., Hoffman D.G., Rees R., 10.1016/0097-3165(92)90099-G, J. Combin. Theory Ser. A 59 (1992), 73–89. Zbl0759.05012MR1141323DOI10.1016/0097-3165(92)90099-G
- Drápal A., Kozlik A., Griggs T.S., 10.1016/j.disc.2011.04.025, Discrete Math. 312 (2012), 597–607. Zbl1321.05021MR2854805DOI10.1016/j.disc.2011.04.025
- Drápal A., Griggs T.S., Kozlik A.R., Basics of DTS quasigroups: Algebra, geometry and enumeration, J. Algebra Appl. 14 (2015), 1550089. Zbl1312.05025MR3338085
- Drápal A., Kozlik A.R., Griggs T.S., Flexible Latin directed triple systems, Utilitas Math.(to appear).
- Ge G., Group divisible designs, Handbook of Combinatorial Designs, second edition, ed. C.J. Colbourn and J.H. Dinitz, Chapman and Hall/CRC Press, Boca Raton, FL, 2007, pp. 255–260. MR2246267
- Ge G., Ling A.C.H., 10.1016/j.disc.2004.04.003, Discrete Math. 285 (2004), 97–120. MR2074843DOI10.1016/j.disc.2004.04.003
- Ge G., Rees R.S., On group-divisible designs with block size four and group-type , Discrete Math. 279 (2004), 247–265. MR2059993
- Ge G., Rees R., Zhu L., 10.1006/jcta.2001.3246, J. Combin. Theory Ser. A 98 (2002), 357–376. MR1899631DOI10.1006/jcta.2001.3246
- Hanani H., 10.1016/0012-365X(75)90040-0, Discrete Math. 11 (1975), 255–369. Zbl0361.62067MR0382030DOI10.1016/0012-365X(75)90040-0
- Kozlik A.R., Cyclic and rotational Latin hybrid triple systems, submitted.
- McCune W., Mace Reference Manual and Guide, Tech. Memo ANL/MCS-TM-264, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, August 2003.
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