Groebner bases for algebraic systems and homology, homotopy

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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