Groebner bases for algebraic systems and homology, homotopy

近年来，由有限个生成元和关系定义的代数系统（拟表示代数系统）上的各种判定问题由于与计算理论的关系而得到了广泛的研究，本文从重写系统的角度研究了拟表示代数系统，特别是拟表示幺半群和结合代数。研究了代数系统具有有限完全重写系统的条件以及与同调同伦的关系，如果幺半群具有有限完全重写系统，则幺半群满足同调有限性FP 3和同伦有限性FDT（Squier）.骄傲介绍了另一个同调有限性性质FHT，并表明，FHT如下FDT。在第三篇文献中，我们证明了有限表示幺半群的FHT等价于bi-FP 3。众所周知，有限表示幺半群的许多性质是不可判定的。我们公关 关于我们 在第一篇文章中，我们证明了它们是不可判定的，即使是在线性时间内具有可判定的字问题的双表示幺半群。此外，在第二篇论文中，我们证明了么半群的许多同调性质也是不可判定的。关于群和群代数的中心的不可判定性的结果将在即将发表的论文（第六篇）中发表。在第五篇论文中，我们从重写系统的角度发展了一般结合代数及其自由双模上的Groebner基理论，并将其应用于计算代数的Hochschild上同调。这是本研究的主要成果，我们期望将其应用于更普遍的2002年2月在京都数学科学研究所、12月在神奈川工业大学、12月在日本东京大学数学系、12月在东京大学数学系、12月2003年在东邦大学举行了关于代数系统和计算的研究会议。结果汇编成一份研究报告中的第七篇文章在参考文献减

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