关于岩浆幺半群的变换半群视角
Transformation Semigroup Perspective on the Magma Monoid
AI总结：
本文从变换半群视角刻画岩浆幺半群的主左、右理想、幂等元和正则元并枚举，为相关理想结果提供通用框架，还解答了开放性问题并纠正前人错误。
AI中文摘要：
2008年H. S. Kim和J. Neggers首次引入了所有二元运算的幺半群。此后，人们研究了该幺半群的不同方面和应用，但关于其半群理论性质的几个问题仍未得到解答。我们采用变换半群视角来全面刻画该幺半群的主左、右理想、幂等元和正则元，并给出它们精确的组合枚举。此方法为岩浆幺半群中关于理想的大多数现有结果提供了一个通用框架。我们还回答了A. Rafieipour在2023年博士论文中提出的几个开放性问题。最后，我们纠正了H. F. Fayomi 2011年论文中关于岩浆幺半群中心描述的一个错误。
英文摘要：
The monoid of all binary operations was first introduced by H. S. Kim and J. Neggers in 2008. Since then, different aspects and applications of this monoid were studied, while several questions about its semigroup-theoretic properties remain unanswered. We employ a transformation semigroup perspective to fully characterize principal left and right ideals, idempotent and regular elements of this monoid, as well as provide precise combinatorial enumerations of them. This approach gives a general framework for most of the existing results on ideals in the magma monoid. We also answer several open questions posed in the 2023 PhD dissertation of A. Rafieipour. Finally, we correct an error regarding the description of the center of the magma monoid from the 2011 paper of H. F. Fayomi.
