{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:09:33Z","timestamp":1753884573390,"version":"3.41.2"},"reference-count":32,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","funder":[{"name":"the Polish National Agency for Academic Exchange and the National Science Centre","award":["2016\/21\/N\/ST1\/01465","2021\/43\/B\/ST1\/00405"],"award-info":[{"award-number":["2016\/21\/N\/ST1\/01465","2021\/43\/B\/ST1\/00405"]}]},{"name":"the Narodowe Centrum Nauki","award":["2016\/22\/E\/ST1\/00450"],"award-info":[{"award-number":["2016\/22\/E\/ST1\/00450"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:p>We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the \u201cWeak Independence Theorem\u201d for pseudo-algebraically closed (PAC) substructures of an ambient structure with no finite cover property (nfcp) and the property [Formula: see text]. Fourth, we describe Kim-dividing in these PAC substructures and show several results related to the SOP<jats:sub>n<\/jats:sub>hierarchy. Fifth, we characterize the algebraic closure in PAC structures.<\/jats:p>","DOI":"10.1142\/s0219061322500301","type":"journal-article","created":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T08:25:23Z","timestamp":1667982323000},"source":"Crossref","is-referenced-by-count":0,"title":["Co-theory of sorted profinite groups for PAC structures"],"prefix":"10.1142","volume":"23","author":[{"given":"Daniel Max","family":"Hoffmann","sequence":"first","affiliation":[{"name":"Instytut Matematyki, Uniwersytet Warszawski, Warszawa, Poland"}]},{"given":"Junguk","family":"Lee","sequence":"additional","affiliation":[{"name":"Center for Mathematical Challenges, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, 02455 Seoul, South Korea"}]}],"member":"219","published-online":{"date-parts":[[2023,1,5]]},"reference":[{"issue":"2","key":"S0219061322500301BIB001","doi-asserted-by":"crossref","first-page":"161","DOI":"10.2307\/1970438","volume":"85","author":"Ax J.","year":"1967","journal-title":"Ann. 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