arXiv subjects
Publications and source records attributed to Andrew R. Kozlik.
A directed triple system can be defined as a decomposition of a complete digraph to directed triples $\langle x,y,z\rangle$. By setting $xy =z$, $yz =x$, $xz =y$ and $uu =u$ we get a binary operation that can be a quasigroup. We give an algebraic description of such quasigroups, explain how they can be associated with triangulated pseudosurfaces and report enumeration results.
It is well known that, given a Steiner triple system, a quasigroup can be formed by defining an operation $\cdot$ by the identities $x \cdot x = x$ and $x \cdot y = z$ where $z$ is the third point in the block containing the pair $\{x,y\}$. The same is true for a Mendelsohn triple system where the pair $(x,y)$ is considered to be ordered. But it is not true in general for directed triple systems. However directed triple systems which form quasigroups under this operation do exist and we call these Latin directed triple systems. The quasigroups associated with Steiner and Mendelsohn triple systems satisfy the flexible law $x \cdot (y \cdot x) = (x \cdot y) \cdot x$ but those associated with Latin directed triple systems need not. In a previous paper, [Discrete Mathematics 312 (2012), 597-607], we studied non-flexible Latin directed triple systems. In this paper we turn our attention to flexible Latin directed triple systems.