发表机构
Chongqing University of Technology（重庆理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探讨满足$x^n\boldsymbol{\text{≈}}x$的环簇$\boldsymbol{R}_n$与加法可交换幂等半环簇$\boldsymbol{W}$的Mal'cev乘积，证明其等式关系、次直不可约元特征等结论，并推广Wang和Shao的相关定理。
AI 中文摘要
设$\boldsymbol{R}_n$为满足$x^n=x$（其中$n\boldsymbol{\text{≥}}2$）的环簇，$\boldsymbol{W}$为加法可交换的幂等半环簇。等式$\boldsymbol{R}_n\boldsymbol{\text{∘}}\boldsymbol{W}=\boldsymbol{R}_n\boldsymbol{\text{∨}}\boldsymbol{W}$成立，且该簇的子簇格为$L(\boldsymbol{R}_n)\boldsymbol{\text{×}}L(\boldsymbol{W})$。其非加法幂等的次直不可约元恰有一个非平凡环分量，该分量为有限域；此类元由该有限域与带乘法单位元的幂等半环确定，后者的一元多项式函数上的分离条件刻画了次直不可约性。每个子簇都有有限恒等式基，一个四元半环生成的簇具有基数无界的次直不可约元。这些结果将Wang和Shao的Mal'cev乘积定理从吸收子簇推广到任意$\boldsymbol{W}\boldsymbol{\text{≤}}\boldsymbol{Slp}$。
英文摘要
Let $\Rn$ be the variety of rings satisfying $x^n\eqid x$, where $n\geq2$, and let $\W$ be a variety of idempotent semirings with commutative addition. The equality $\Rn\circ\W=\Rn\vee\W$ holds, and this variety has subvariety lattice $L(\Rn)\times L(\W)$. Its subdirectly irreducible members that are not additively idempotent have exactly one nontrivial ring component, which is a finite field. Such a member is determined by this field and an idempotent semiring with a multiplicative identity; a separation condition on unary polynomial functions of the latter characterizes subdirect irreducibility. Every subvariety has a finite identity basis. A four-element semiring generates a variety with subdirectly irreducible members of unbounded cardinality. The results extend the Mal'cev product theorem of Wang and Shao from the absorption subvariety to arbitrary $\W\leq\Slp$.
