# Critères de finitude homologique pour la non convergence des systèmes de réécriture de termes

Abstract : We construct homological finiteness conditions for the existence of finite type convergent presentations by rewriting of one-sorted and first-order equational theories. An equational theory is semantically described by an algebraic theory in the sense of Lawvere. Generalising the MacLane homology of rings, we introduce the homology of such an algebraic theory with coefficients in non additive bimodules. This homology can be interpretated in terms of Hochschild-Mitchell homology of the underlying small category. We generalise the free resolutions of Squier and Kobayashi from the string rewriting to the rewriting of morphisms in small categories. By using these resolutions, we prove that any algebraic theory admitting a finite type convergent presentation is of type bi-$\mathrm(PF)_(\infty)$. We construct a decidable non unitary equational theory which does not admit a finite type convergent presentation.
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https://tel.archives-ouvertes.fr/tel-00008784
Contributor : Philippe Malbos <>
Submitted on : Tuesday, March 15, 2005 - 1:22:54 PM
Last modification on : Thursday, January 11, 2018 - 6:15:40 AM
Long-term archiving on: : Friday, April 2, 2010 - 10:04:10 PM

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• HAL Id : tel-00008784, version 1

### Citation

Philippe Malbos. Critères de finitude homologique pour la non convergence des systèmes de réécriture de termes. Mathématiques [math]. Université Montpellier II - Sciences et Techniques du Languedoc, 2004. Français. ⟨tel-00008784⟩

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