Existence of extensions of semilattices of groups by groups, cohomology, and crossed modules for inverse semigroups

• Autor(a) principal: Makuta, Thaís Mayumi Batista
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-16062020-172746/
• topic: Abstract kernel; Ação parcial; Cohomologia de ação parcial; Cohomologia de semigrupos inversos; Extensions; Extensões; Inverse semigroup cohomology; Inverse semigroups; Núcleo abstrato; Partial action; Partial group cohomology; Semigrupos inversos; Semilattices of groups; Semirreticulados de grupos

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.