Rewriting systems (Groebner bases) on algebraic systems and their application

基于代数系统的重写系统（Groebner 基础）及其应用

• 批准号: 17540042
• 负责人: KOBAYASHI Yuji
• 金额: $ 1.06万
• 依托单位: Toho University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540042/
• 关键词: finitetly presented algebra; rewriting systems; Groebner basis; cohomologv; roiective resolution; cup product; homological finiteness; undecidability; ジグザグ; Groebner基底; アルゴリズム; Hochschildコホモロジー; syzygy; 組合せデザイン; 決定問題

Complete rewriting systems and Groebner bases give effective tools to solve algorithmical problems on algebraic systems and have been studied intensively.In the present research, we study finitely presented algebraic systems (algebraic systems defined by a finite number of generators and a finite number of relations), particularly, monoids and associative algebras by means of rewriting systems (Groebner bases). We develop a unified theory by treating Groebner bases as rewriting systems on additive groups We formulate a notion of critical pairs in this situation and clarify the role of them in the theory.Based on the theory of Groebner bases on associative algebras and projective modules on them, we construct projective resolutions, and develop the methods to compute the Hochschild cohomology. It makes possible to not only compute cohomology but also determine the ring structure of it by giving explicitly the cup products of cocycles.Moreover, we study finiteness of low dimensional cohomology. It is known since Squier that monoids has homological finiteness property FPn in every dimension n if they have complete rewriting systems. The finiteness for dimension 2 is related to the finite presentability of monoids, but details are not known. In this research, we study the one dimensional case and find that the finiteness of 1-dimensional cohomology is related to the finite generation and zigzags of monoidsMany properties of finitely presented monoids and groups are undecidable. In this research, we show that the triviality of the centers of monoids and groups are undecidable. This result is of interest because it means that even the 0-dimensional cohomology is not computable in general.

完整的重写系统和Groebner基础提供了有效的工具来解决代数系统上的算法问题，并已进行了深入研究。在本研究中，我们研究了有限呈现的代数系统（代数系统（代数系统，由有限数量的生成器和有限的关系定义），尤其是Monicoids and Assopiative Algebras Systems（Rebrite andrient）（Rebrient）（Rebrient）（Rebrient）（Rebrient）（Rebrient）（Rebrient）（我们通过将Groebner基础视为添加剂组的重写系统来发展统一的理论，在这种情况下，我们制定了关键对的概念，并阐明了它们在理论中的作用。基于Groebner理论基于关联代数及其对它们的投射模块的基础，我们对它们的构建投影解决方案进行了构建和制定方法，以构建Hochschschschschorlogy cohomology cohomology cohomology。不仅可以通过明确给出同伴的杯产物来确定其环结构。此外，我们还研究了低维共同体学的有限性。众所周知，Squier是Monoids具有完整的重写系统，在每个维度N中都具有同源性属性fpn。维度2的有限性与单粒细胞的有限呈现性有关，但细节尚不清楚。在这项研究中，我们研究了一个维度的情况，发现一维共同体的有限性与有限呈现的单体和基团有限的单体特性的有限产生和锯齿形有关。在这项研究中，我们表明，单体和群体中心的微不足道是不可确定的。该结果引起了人们的关注，因为这意味着即使是0维的共同体也是一般的计算。

项目成果

Cluttered orderings for the complete bipartite graph

• DOI: 10.1016/j.dam.2005.06.005
• 发表时间: 2005-11
• 期刊: Discret. Appl. Math.
• 作者: Meinard Müller;T. Adachi;Masakazu Jimbo

Combinatorial structure of group divisible designs and their costructions

群可分设计的组合结构及其构造

• 发表时间: 2005
• 期刊: 京都大学数理解析研究所講究録 1437
• 作者: Y.;Kobayashi;T.;Adachi (ed);Yuji Kobayashi;Yuji Kobayashi;Tomoko Adachi;足立智子;Yuji Kobayashi;Tomoko Adachi
• 通讯作者: Tomoko Adachi

Labelings for the complete bitartite graph and its applications

完整的bitartite图的标记及其应用

• 发表时间: 2007
• 期刊: 京都大学数理解析研究所講究録 1562
• 作者: Tomoko;Adachi;Yuji kobayashi;Tomoko Adachi
• 通讯作者: Tomoko Adachi

Rewriting systems, Grobner bases and syzygies on algebras and modules

重写系统、Grobner 基础以及代数和模的 syzygies

• 发表时间: 2007
• 作者: Yuji;Kobayashi;Yuji Kobayashi
• 通讯作者: Yuji Kobayashi

Proc.of Algebra, Languages and Computation

代数、语言与计算学进展

• 发表时间: 2006
• 作者: Y. Kobayashi;T. Adachi(ed.)
• 通讯作者: T. Adachi(ed.)