Groebner bases for algebraic systems and homology, homotopy

近年来，由于与计算理论的关系，各种由有限数量的生成器和关系所定义的代数系统（有限呈现代数系统）上的决策问题得到了广泛的研究。在本研究中，我们研究了有限表示的代数系统，特别是有限表示的单群和结合代数的重写系统。研究了代数系统具有有限完全改写系统的条件及其与同调、同伦的关系。如果一个单群具有有限完全改写系统，则它满足同列有限性FP3和同列有限性FDT（由Squier提出）。Pride引入了FHT的另一个同构有限性，并证明了FHT是由FDT衍生出来的。在参考文献的第三篇论文中，我们证明了FHT等价于bi-FP3。众所周知，有限单群上的许多性质是（递归地）不可确定的。在第一篇论文中我们发现，即使对于线性时间内可判定的有限单群，它们也是不可判定的。此外，在第二篇论文中，我们还证明了一元群的许多同调性质也是不可确定的。关于群和群代数中心的不可判定性的结果将在即将发表的论文（第六篇论文）中发表。在第五篇论文中，我们从改写系统的观点出发，发展了基于一般结合代数及其自由双模的Groebner理论，并将其应用于代数的Hochschild上同调的计算。这是本研究的主要成果，我们期望将其应用于更一般（或特殊）的代数系统。在第四篇论文中，我们给出了一种通过恩里克格来研究恩里克曲面的方法。2002年2月在京都数学科学研究所，12月在神奈川工业大学，2003年12月在东宝大学，我们举行了关于代数系统和计算的研究会议。研究结果被汇编成一份研究报告，发表在《参考文献》杂志的第七篇文章中

Recently, various decision problems on algebraic systems defined by a finite number of generators and relations (finitely presented algebraic systems) have been studied extensively because of relationship to computation theory.In this research, we studied finitely presented algebraic systems, in particular, finitely presented monoids and associative algebras in terms of rewriting systems. We investigated conditions for algebraic systems to have finite complete rewriting systems and relationship to homology and homotopy.If a monoid has a finite complete rewriting system, then it satisfies the homological finiteness property FP3 and the homotopical finiteness property FDT (by Squier). Pride introduced another homological finiteness property FHT and showed that FHT follows from FDT. We showed that FHT is equivalent to bi-FP3 for finite presented monoids in the third paper in REFERENCES.It is well-known that many properties on finitely presented monoids are (recursively) undecidable. We pr … More oved in the first paper that they are undecidable even for finitely presented monoids with word problem decidable in linear time. Moreover, in the second paper we showed that many homological properties of monoids are also undecidable. Results about the undecidability of the centers of groups and group algebras will be published in a forthcoming paper (the sixth paper).In the fifth paper we developed the theory of Groebner bases on general associative algebras and their free bimodules from a viewpoint of rewriting systems and applied it to compute the Hochschild cohomology of algebras. This is the main results of this research, and we expect to apply them to more general (or special) algebraic systems.In the forth paper we gave a method to study Enriques surfaces via Enriques lattices.In February, 2002 at Kyoto Research Institute of Mathematical Science, in December at Kanagawa Institute of Technology, and in December, 2003 at Toho University we hold research meetings on algebraic systems and computations. The results are compiled into a research report in the seventh article in REFERENCES Less