有限半群与半群簇

Finite semigroups and semigroup varieties are not only the important research contents of semigroup theory and algebra variety theory, but also have been widely applied to information science, theoretical computer science, formal language theory, automata theory, symbolic dynamics, discrete mathematics, graph theory, cryptography and other disciplines. This program will do some researches around finite semigroups and semigroup varieties, various finite basis problems for some important finite semigroups, unary semigroups and discrete syntactic monoids of languages will be studied, and then the general solutions to solve the finite basis problem for finite semigroups will be explored; the subvarieties and the structures of its subvariety lattices for some important semigroup varieties will be characterized, all of minimal finitely based semigroups that generate non-small varieties will be determined, and varieties of monoids which have modular and distributive lattices of subvarieties will be characterized; the problem of the computational complexity of Var-Memb for some important nonfinitely based semigroups will be studied and solved, and the lower bound for the computational complexity of the finite basis problem for finite semigroups will be established. These problems are the focus in the recent research. The research of this scheme will enrich the research contents of the theory of semigroups and the theory of algebra varieties, which has not only important theoretical significance, but also very good application prospects.

有限半群和半群簇不仅是半群代数理论和代数簇理论的重要研究内容，而且在信息科学、理论计算机科学、形式语言理论、自动机理论、符号动力学、离散数学、图论、密码学等学科中都有广泛的应用。本项目计划围绕有限半群和半群簇开展工作，研究一些重要的有限半群、酉半群和语言的离散句法幺半群的各种有限基问题，探索解决有限半群的有限基问题的一般方法；刻画一些重要半群簇的子簇及其子簇格的结构，确定所有极小 non-small 有限基半群，刻画具有模子簇格和分配子簇格的幺半群簇；研究和解决一些重要非有限基半群的 Var-Memb 算法复杂性问题，确定有限半群的有限基问题的算法复杂性下界。这些问题都是本领域目前研究的热点问题。本项目的研究将丰富半群代数理论和代数簇理论的研究内容，不仅具有重要的理论意义，而且还有很好的应用前景。

有限半群和半群簇不仅是半群代数理论和代数簇理论的重要研究内容，而且在信息科学、理论计算机科学、形式语言理论、自动机理论、符号动力学、离散数学、图论、密码学等学科中都有广泛的应用。本项目围绕有限半群和半群簇开展工作，给出了一些判定半群是有限基和非有限基的充分条件，解决了一些重要的有限半群、离散句法幺半群、对合半群、矩阵半群、变换半群、Kiselman幺半群的各种有限基问题；刻画了一些重要半群簇的子簇及其子簇格的结构，找到了一个limit簇；研究和解决了一些重要非有限基半群的 Var-Memb算法复杂性问题；研究了丛代数和表示论的相关内容。这些问题都是本领域目前研究的热点问题。本项目的研究丰富了半群代数理论和代数簇理论的研究内容，不仅具有重要的理论意义，而且还有很好的应用前景。