AI 中文总结
该研究建立有限半群完全映射理论，证明存在完全映射的有限半群为正则且问题可归约为主因子，分类多类半群的完全映射存在性，给出典型半群的存在性结论并提出开放问题。
AI 中文摘要
半群$S$的完全映射是指一个双射$\alpha\colon S\to S$，使得由$x\theta=x\cdot x\alpha$定义的映射$\theta\colon S\to S$同样是双射。等价地，它确定了$S$乘法表的一个横截。完全映射连接了群论、拉丁方与密码学，其在有限群中的存在性已通过Hall-Paige猜想的证明得到刻画。本文建立了有限半群的相应完全映射理论。\n 我们证明了所有存在完全映射的有限半群都是正则的，且该问题可归约为主因子问题。我们对无零Rees矩阵半群中完全映射的存在性进行了分类，给出了具有完全映射的群上的Rees 0-矩阵半群的Hall型判别准则，并证明了极大子群不具有完全映射的Rees 0-矩阵半群存在完全映射的充分条件。作为Rees 0-矩阵分析的主要应用，我们证明了全变换半群$T_n$存在完全映射当且仅当$n=1$或$n\geq4$，等价地，$T_n$存在完全映射当且仅当对称群$S_n$也存在完全映射。我们还证明了有限维向量空间的全线性幺半群存在完全映射，仅在奇序域上的1维情形和二元域$\mathbb F_2$上的2维情形除外。此外我们证明了分划幺半群$\mathcal P_n$存在完全映射当且仅当$n=1$或$n\ge4$，且所有有限无周期正则$*$-半群都存在完全映射，由此可推出平面分划幺半群、Motzkin幺半群和Jones幺半群都存在完全映射。\n 本文最后提出了若干开放问题。
英文摘要
A complete mapping of a semigroup $S$ is a bijection $α\colon S\to S$ such that the map $θ\colon S\to S$ defined by $xθ=x\cdot xα$ is also a bijection. Equivalently, it determines a transversal of the multiplication table of $S$. Complete mappings connect group theory, Latin squares, and cryptography, and their existence for finite groups was characterized by the resolution of the Hall--Paige conjecture. In this paper, we develop the corresponding theory for finite semigroups. We prove that every finite semigroup admitting a complete mapping is regular and that the problem reduces to principal factors. We classify the existence of a complete mapping in Rees matrix semigroups without zero, give a Hall-type criterion for Rees $0$-matrix semigroups over groups with complete mappings, and prove sufficient conditions for Rees $0$-matrix semigroups whose maximal subgroups do not have complete mappings. As the main application of the Rees $0$-matrix analysis, we show that $T_n$ has a complete mapping if and only if $n=1$ or $n\geq 4$. Equivalently, $T_n$ has a complete mapping if and only if the same holds for $S_n$. We prove that the full linear monoid of a finite-dimensional vector space has a complete mapping except in dimension $1$ over a field of odd order and in dimension $2$ over $\mathbb F_2$. We also prove that the partition monoid $\mathcal P_n$ has a complete mapping if and only if $n=1$ or $n\ge4$, and that every finite aperiodic regular $*$-semigroup has a complete mapping. As a consequence, the planar partition, Motzkin and Jones monoids have complete mappings. The paper concludes with open problems.
Comments63 pages
