Study of the word problem for algebraic systems by means of rewriting

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We studied the word problem and other decision problems for finitely presented algebras by means of rewriting systems.We found some relationship between the solvability of the word problem and the existence of complete rewriting systems with good properties in a language- theoretical sense for finitely presented monoids. We also studied cross-sections of monoids related to the above properties. For the word problem to be solvable, context-sensitive cross-sections suffice but context-free cross-sections do not (see [2] and [9]). We reported these results in a survey article [5].We studied some important properties such as confluence and termination of rewriting systems themselves. In [6] we gave a result on the termination for confluent one-rule systems.We showed that the rewriting techniques are useful too in the homotopy theory of the derivation graphs associated with monoid presentations. If a monoid has a complete homotopy reduction system, then it satisfies the homological finiteness property FP4. We always have the left canonical reduction system and it is complete if the presentation is nonspecial. These results are reported in [8].We developed the method to construct a family of algebraic curves of genus g <greater than or equal> 2 with large rank modifying Neron's method