AI 中文总结
本文研究半对称拟群的核，证明其核的结构特征，推导特定阶半对称拟群存在的充要条件，并刻画门德尔松环的核元素，丰富了拟群与环的结构理论。
AI 中文摘要
满足恒等式$(x \cdot y) \cdot x = y$的二元运算$\boldsymbol{\text{semisymmetric quasigroup}}$（半对称拟群），其核要么为空，要么是与中心重合的初等阿贝尔2群；非空核的半对称拟群必为$\boldsymbol{\text{Mendelsohn loop}}$（门德尔松环，即与门德尔松三元组系统关联的环）。本文推导了阶为$n$且核阶为$m$的半对称拟群存在的充要条件，并通过关联三元组系统中帕施构形的特定定向，刻画了门德尔松环的核元素。
英文摘要
A binary operation $\cdot$ which satisfies the identity $(x \cdot y) \cdot x = y$ is called a semisymmetric quasigroup. We show that the nucleus of a semisymmetric quasigroup is either empty or an elementary abelian 2-group coinciding with the centre, and that a semisymmetric quasigroup with a non-empty nucleus is necessarily a Mendelsohn loop, i.e. the loop associated with a Mendelsohn triple system. We derive necessary and sufficient conditions for the existence of a semisymmetric quasigroup of order $n$ with nucleus of order $m$. Furthermore, we characterize the nuclear elements of a Mendelsohn loop in terms of a particular orientation of the Pasch configuration in the associated triple system.
Commentsv2: simplified proof of Theorem 3.2, added acknowledgement
