Basics of DTS quasigroups: algebra, geometry and enumeration

Abstract

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.