An important problem in universal algebra consists in finding#R##N#presentations of algebraic theories by generators and relations, which#R##N#are as small as possible. Exhibiting lower bounds on the number of#R##N#those generators and relations for a given theory is a difficult task#R##N#because it a priori requires considering all possible sets of#R##N#generators for a theory and no general method exists. In this article,#R##N#we explain how homological computations can provide such lower bounds,#R##N#in a systematic way, and show how to actually compute those in the#R##N#case where a presentation of the theory by a convergent rewriting#R##N#system is known. We also introduce the notion of coherent presentation#R##N#of a theory in order to consider finer homotopical invariants. In some#R##N#aspects, this work generalizes, to term rewriting systems, Squier's#R##N#celebrated homological and homotopical invariants for string rewriting#R##N#systems.
['Philippe Malbos', 'Samuel Mimram']
Homological Computations for Term Rewriting Systems