Derivations and relation modules for inverse semigroups

N. D. Gilbert


We define the derivation module for a homomorphism of inverse semigroups, generalizing a construction for groups due to Crowell.  For a presentation map from a free inverse semigroup, we can then define its relation module as the kernel of a canonical map from the derivation module to the augmentation module.  The constructions are analogues of the first steps in the Gruenberg resolution obtained from a group presentation.  We give a new proof of the characterization of inverse monoids of cohomological dimension zero, and find a class of examples of inverse semigroups of cohomological dimension one.


inverse semigroup, cohomology, derivation, relation module

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