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word rewriting, Gröbner bases, homology of algebras and monoids. Scientific context. Since the eighties, several results (Anick [1], Squier [11], Kobayashi [7], Brown [2], Dotsenko [4, 5], etc.) have proved that rewriting methods can be used to solve algebraic decision problems and to construct homological invariants. Indeed, the combinatorial study of groups, of monoidal structures (monoids, categories, pros), of cartesian structures (Lawvere theories [8]) or of linear structures (associative algebras, operads) is founded on presentations by generators and relations with “good ” computational properties (finiteness, termination, confluence). In particular, convergent presentations (terminating and confluent) and Gröbner bases (for linear structures, [3]) can compute normal forms and, as a consequence, they can solve decision problems (Dehn problems, combinatorial problems for Artin-Tits groups, etc.) by rewriting methods. More generally, several algorithmic methods for computing free Abelian resolutions from convergent presentations or Gröbner bases have been developed (Anick, Kobayashi, etc.) to allow the explicit computation of homological invariants. Subject. The presentations of algebras generalise the notion of rewriting systems to linear combinations of words. Given an associative algebra A on a field K, a method to compute its homolog

Year: 2013

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