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Doctoral Thesis
DOI
https://doi.org/10.11606/T.45.2020.tde-16062020-172746
Document
Author
Full name
Thaís Mayumi Batista Makuta
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2020
Supervisor
Committee
Dokuchaev, Mikhailo (President)
Alves, Marcelo Muniz Silva
Kochloukova, Dessislava Hristova
Marcos, Eduardo do Nascimento
Paques, Antonio
Title in English
Existence of extensions of semilattices of groups by groups, cohomology, and crossed modules for inverse semigroups
Keywords in English
Abstract kernel
Extensions
Inverse semigroup cohomology
Inverse semigroups
Partial action
Partial group cohomology
Semilattices of groups
Abstract in English
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.
Title in Portuguese
Existência de extensões de semirreticulados de grupos por grupos, cohomologia, e módulos cruzados para semigrupos inversos
Keywords in Portuguese
Ação parcial
Cohomologia de ação parcial
Cohomologia de semigrupos inversos
Extensões
Núcleo abstrato
Semigrupos inversos
Semirreticulados de grupos
Abstract in Portuguese
Introduz-se o conceito de um núcleo abstrato parcial associado a um par (G,A), em que G é um grupo e A é um semirreticulado de grupos, e relaciona-se o grupo de cohomologia parcial H^3(G,C(A)) às obstruções a existência de extensões admissíveis de A por G que realizam o núcleo abstrato dado. Também, se tais extensões existem, mostra-se que elas são classificadas por H^2(G,C(A)). Define-se a noção de módulo cruzado sobre um semigrupo inverso e constrói-se uma sequência de quatro termos correspondente. A cada classe de equivalência de tais sequências relaciona-se um elemento do terceiro grupo das cohomologias que perservam ordem, que no caso de semirreticulado de grupos resulta numa bijeção.
 
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Publishing Date
2021-01-20
 
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