{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T12:12:40Z","timestamp":1772367160446,"version":"3.50.1"},"reference-count":19,"publisher":"Walter de Gruyter GmbH","issue":"5","funder":[{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u01ceo de Amparo \u00e0 Pesquisa do Estado de S\u01ceo Paulo","doi-asserted-by":"publisher","award":["2015\/09162-9"],"award-info":[{"award-number":["2015\/09162-9"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["307873\/2017-0"],"award-info":[{"award-number":["307873\/2017-0"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e 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Tecnologia","doi-asserted-by":"publisher","award":["PTDC\/MAT-PUR\/31174\/2017"],"award-info":[{"award-number":["PTDC\/MAT-PUR\/31174\/2017"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We introduce the concept of a partial abstract kernel associated to a group <jats:italic>G<\/jats:italic> and a semilattice of groups <jats:italic>A<\/jats:italic> and relate the partial cohomology group <jats:inline-formula id=\"j_forum-2019-0281_ineq_9999_w2aab3b7e4348b1b6b1aab1c15b1b5Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:msup>\n                                 <m:mi>H<\/m:mi>\n                                 <m:mn>3<\/m:mn>\n                              <\/m:msup>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>G<\/m:mi>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mrow>\n                                    <m:mi>C<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mrow>\n                                       <m:mo>(<\/m:mo>\n                                       <m:mi>A<\/m:mi>\n                                       <m:mo>)<\/m:mo>\n                                    <\/m:mrow>\n                                 <\/m:mrow>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forum-2019-0281_eq_0249.png\"\/>\n                        <jats:tex-math>{H^{3}(G,C(A))}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> with the obstructions to the existence of admissible extensions of <jats:italic>A<\/jats:italic> by <jats:italic>G<\/jats:italic> which realize the given abstract kernel.\nWe also show that if such extensions exist, then they are classified by <jats:inline-formula id=\"j_forum-2019-0281_ineq_9998_w2aab3b7e4348b1b6b1aab1c15b1c11Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:msup>\n                                 <m:mi>H<\/m:mi>\n                                 <m:mn>2<\/m:mn>\n                              <\/m:msup>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>G<\/m:mi>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mrow>\n                                    <m:mi>C<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mrow>\n                                       <m:mo>(<\/m:mo>\n                                       <m:mi>A<\/m:mi>\n                                       <m:mo>)<\/m:mo>\n                                    <\/m:mrow>\n                                 <\/m:mrow>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forum-2019-0281_eq_0248.png\"\/>\n                        <jats:tex-math>{H^{2}(G,C(A))}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1515\/forum-2019-0281","type":"journal-article","created":{"date-parts":[[2020,9,7]],"date-time":"2020-09-07T12:37:48Z","timestamp":1599482268000},"page":"1297-1313","source":"Crossref","is-referenced-by-count":1,"title":["The third partial cohomology group and existence of extensions of semilattices of groups by groups"],"prefix":"10.1515","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1250-4831","authenticated-orcid":false,"given":"Mikhailo","family":"Dokuchaev","sequence":"first","affiliation":[{"name":"Instituto de Matem\u00e1tica e Estat\u00edstica , Universidade de S\u00e3o Paulo , Rua do Mat\u00e3o, 1010 , S\u00e3o Paulo , SP, CEP: 05508\u2013090 , Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4504-3261","authenticated-orcid":false,"given":"Mykola","family":"Khrypchenko","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica , Universidade Federal de Santa Catarina , Campus Reitor Jo\u00e3o David Ferreira Lima , Florian\u00f3polis , SC, CEP: 88040\u2013900 , Brazil ; and Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es, Faculdade de Ci\u00eancias e Tecnologia, Universidade Nova de Lisboa, 2829\u2013516 Caparica, Portugal"}]},{"given":"Mayumi","family":"Makuta","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica e Estat\u00edstica , Universidade de S\u00e3o Paulo , Rua do Mat\u00e3o, 1010 , S\u00e3o Paulo , SP, CEP: 05508\u2013090 , Brazil"}]}],"member":"374","published-online":{"date-parts":[[2020,5,9]]},"reference":[{"key":"2023033119395826080_j_forum-2019-0281_ref_001_w2aab3b7e4348b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"E.  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