### Abstract

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.

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.

Original language | English |
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Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | Algebra and Discrete Mathematics |

Volume | 12 |

Issue number | 1 |

Publication status | Published - 2011 |

### Keywords

- Inverse semigroup
- cohomology
- derivation
- relation module