Inverse semigroup cohomology and crossed module extensions of semilattices of groups by inverse semigroups

TL;DR

This paper introduces the concept of crossed modules over inverse semigroups and establishes a cohomological classification of their extensions, linking algebraic structures with cohomology groups.

Contribution

It defines crossed modules over inverse semigroups and proves a correspondence between their extensions and third cohomology groups, advancing the understanding of inverse semigroup cohomology.

Findings

Classification of crossed module extensions via cohomology

Establishment of a one-to-one correspondence with $H^3_ le(T^1,A^1)$

Development of a framework connecting inverse semigroup theory and cohomology

Abstract

We define and study the notion of a crossed module over an inverse semigroup and the corresponding $4$-term exact sequences, called crossed module extensions. For a crossed module $A$ over an $F$-inverse monoid $T$, we show that equivalence classes of admissible crossed module extensions of $A$ by $T$ are in a one-to-one correspondence with the elements of the cohomology group $H^3_\le(T^1,A^1)$.