Cyclic and rotational Latin hybrid triple systems

Year, pages: 2017, 1055 - 1072

It is well known that given a Steiner triple system (STS) one can define a binary operation $*$ upon its base set by assigning $x*x=x$ for all $x$ and $x*y=z$, where $z$ is the third point in the block containing the pair $\{x,y\}$. The same can be done for Mendelsohn triple systems (MTS), directed triple systems (DTS) as well as hybrid triple systems (HTS), where $(x,y)$ is considered to be ordered. In the case of STSs and MTSs the operation yields a quasigroup, however this is not necessarily the case for DTSs and HTSs. A DTS or an HTS which induces a quasigroup is said to be Latin. In this paper we study Latin DTSs and Latin HTSs which admit a cyclic or a 1-rotational automorphism. We prove the existence spectra for these systems as well as the existence spectra for their pure variants. As a side result we also obtain the existence spectra of pure cyclic and pure 1-rotational MTSs.

Kozlik, A. 2017. Cyclic and rotational Latin hybrid triple systems. In Mathematica Slovaca, vol. 67, no.5, pp. 1055-1072. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0032

Kozlik, A. (2017). Cyclic and rotational Latin hybrid triple systems. Mathematica Slovaca, 67(5), 1055-1072. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0032