AI 中文总结
本文针对一般有向三元系未必构成拟群的问题，在已研究非灵活拉丁有向三元系的基础上，聚焦于灵活拉丁有向三元系展开研究。
AI 中文摘要
众所周知，给定一个斯坦纳三元系，可通过恒等式 $x \cdot x = x$ 和 $x \cdot y = z$（其中 $z$ 是包含对 $\{x,y\block$ 的第三个点）定义运算 $\cdot$ 来形成一个拟群；对于将对 $(x,y)$ 视为有序的门德尔松三元系，情况同样如此。但一般而言，有向三元系并非如此，不过存在在该运算下构成拟群的有向三元系，我们称之为拉丁有向三元系。与斯坦纳和门德尔松三元系相关的拟群满足灵活律 $x \cdot (y \cdot x) = (x \cdot y) \cdot x$，但与拉丁有向三元系相关的拟群不一定满足该定律。在之前的论文[《离散数学》312卷（2012），597-607页]中，我们研究了非灵活拉丁有向三元系；本文则将注意力转向灵活拉丁有向三元系。
英文摘要
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.
Journal refUtil. Math. 104 (2017), 31--46
