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directed triple system; quasigroup
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 $\{x,y\}$. 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 $1\pmod{3}$ and $n\geq 13$.
[1] Brouwer A.E., Schrijver A., Hanani H.: Group divisible designs with block-size four. Discrete Math. 20 (1977), 1–10. DOI 10.1016/0012-365X(77)90037-1 | MR 0465894 | Zbl 1093.05008
[2] Colbourn C.J., Hoffman D.G., Rees R.: A new class of group divisible designs with block size three. J. Combin. Theory Ser. A 59 (1992), 73–89. DOI 10.1016/0097-3165(92)90099-G | MR 1141323 | Zbl 0759.05012
[3] Drápal A., Kozlik A., Griggs T.S.: Latin directed triple systems. Discrete Math. 312 (2012), 597–607. DOI 10.1016/j.disc.2011.04.025 | MR 2854805 | Zbl 1321.05021
[4] Drápal A., Griggs T.S., Kozlik A.R.: Basics of DTS quasigroups: Algebra, geometry and enumeration. J. Algebra Appl. 14 (2015), 1550089. MR 3338085 | Zbl 1312.05025
[5] Drápal A., Kozlik A.R., Griggs T.S.: Flexible Latin directed triple systems. Utilitas Math.(to appear).
[6] 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. MR 2246267
[7] Ge G., Ling A.C.H.: Group divisible designs with block size four and group type $g^u m^1$ for small $g$. Discrete Math. 285 (2004), 97–120. DOI 10.1016/j.disc.2004.04.003 | MR 2074843
[8] Ge G., Rees R.S.: On group-divisible designs with block size four and group-type $6^u m^1$. Discrete Math. 279 (2004), 247–265. MR 2059993
[9] Ge G., Rees R., Zhu L.: Group-divisible designs with block size four and group-type $g^u m^1$ with $m$ as large or as small as possible. J. Combin. Theory Ser. A 98 (2002), 357–376. DOI 10.1006/jcta.2001.3246 | MR 1899631
[10] Hanani H.: Balanced incomplete block designs and related designs. Discrete Math. 11 (1975), 255–369. DOI 10.1016/0012-365X(75)90040-0 | MR 0382030 | Zbl 0361.62067
[11] Kozlik A.R.: Cyclic and rotational Latin hybrid triple systems. submitted.
[12] McCune W.: Mace$4$ 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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