Study on the relationship between Green's relationship and cellular structure in the abundant semigroup
富足半群中格林关系与胞腔性结构的关系研究

摘要

富足半群作为半群理论中的重要结构类型，其内部元素在理想生成、局部对称性与幂等分布方面表现出丰富特征。格林关系能够从整体层面描述元素在左右理想意义下的等价性，而胞腔性结构则从局部行为入手，对半群进行更精细的划分。本文围绕二者在富足半群中的相互关系展开讨论，分析格林关系如何为胞腔划分提供结构框架，以及胞腔性结构如何进一步细化格林类内部的代数特征。研究表明，二者结合后可形成更清晰的结构表达，有助于通过多层次视角理解富足半群的整体规律，为进一步探讨半群的局部与全局结构提供参考。

Abstract

As an important structural type in semigroup theory, the abundant semigroup exhibits rich characteristics in terms of ideal generation, local symmetry and idempotent distribution of its internal elements. Green's relation can describe the equivalence of elements in the left-right ideal sense from an overall perspective, while the cellular structure starts from the local behavior and makes a more refined division of the semiggroup. This paper discusses the interrelationship between the two in the abundant semigroup, analyzes how the Green relation provides a structural framework for cell division, and how the cellular structure further refines the algebraic features within the Green class. Research shows that the combination of the two can form a clearer structural expression, which is conducive to understanding the overall law of the abundant semigroup from multiple perspectives and provides a reference for further exploration of the local and global structure of the semigroup.