Existence of extensions of semilattices of groups by groups, cohomology, and crossed modules for inverse semigroups - Makuta, Thaís Mayumi Batista (2020)
We introduce the concept of a partial abstract kernel associated to a pair (G, A), where G is a group and A is a semilattice of groups, and relate the partial cohomology group H^3(G,C(A)) with the obstructions to the existence of admissible extensions of A by G which realize the given abstract kernel. Also, if such extensions exist, we show that they are classified by H^2(G,C(A)). We define the notion of a crossed module over inverse semigroups and construct a corresponding 4-term sequence. To each equivalence class of such sequences we relate an element of the third order-preserving inverse semigroup cohomology, so that we have a bijection in the case of a semilattice of groups.