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

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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.