AI-半环簇的Burnside问题和Tarski问题

The algebraic theory of AI-semirings is one of the most popular subjects in algebra. It has wide applications in theoretical computer science, combinatorial mathematics, functional analysis, topologies, graph theories and dynamic systems. In the last ten years, the research of AI-semiring variety is growing rapidly at home and abroad. In this project, we shall study Burnside problem, restricted Burnside problem, Tarski basis problem and Tarski generation problem for AI-semiring varieties. Firstly, we shall try to establish models of the free objects, characterize subdirectly irreducible members and the subvariety lattice, give the nonfinitely based AI-semirings with the least order and limit AI-semiring varieties, and characterize inherently nonfinitely based AI-semiring varieties. Finally, we shall answer the above questions for specific AI-semiring varieties. This project will further enrich content and research method of semiring algebraic theory and variety theory.

AI-半环的代数理论是代数学中的热点课题之一. 它在理论计算机科学、组合数学、泛函分析、拓扑学、图论和动力系统有着广泛的应用. 近十余年, 国内外关于AI-半环簇的研究方兴未艾. 本项目将围绕Burnside问题, 限制Burnside问题, Tarski基底问题和Tarski生成问题对AI-半环簇展开研究. 努力地建立AI-半环簇的自由对象的模型; 刻画次直不可约成员和子簇格; 给出阶数最小的非有限基底的AI-半环和极限AI-半环簇; 刻画遗传非有限基底的AI-半环簇. 最终, 对若干特定的AI-半环簇就上述几个问题给出解答. 本项目的研究将进一步丰富半环代数理论和簇理论的内容和研究方法.

半环的代数理论和和许多应用领域的问题有着密切联系。我们预定的主要研究的内容是：AI-半环簇的Burnside问题, 限制Burnside问题, Tarski基底问题和Tarski生成问题。本项目大致上按预定计划进行，未作大的调整。部分预定研究目标已取得较好的结果，譬如，在AI-半环簇的子簇格，有限基底问题，有限生成问题等方面，我们取得了较好的研究成果，拓宽了前人的研究工作。本项目的实施进一步丰富了半环代数理论和簇理论的内容和研究方法.