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from Part II - Coherent Presentations
Published online by Cambridge University Press: 18 March 2025
Squier showed in his 1987 article that a convergent presentation of a monoid yields a partial resolution generated by the set of generators in dimension 1, by the set of rules in dimension 2, and by the critical branchings in dimension 3. If moreover the presentation is finite, the Squier resolution is finitely generated up to dimension 3. In this case, the monoid is said to be of homological type left-FP3. This property readily implies that the third integral homology group of the monoid is finitely generated. Therefore, a monoid whose third homology group is not finitely generated does not admit a finite convergent presentation. By explicitly exhibiting an example of this type, Squier first provided a negative answer to the question of universality of convergent rewriting.
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