{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T08:25:52Z","timestamp":1774340752846,"version":"3.50.1"},"reference-count":24,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","funder":[{"name":"KIAS","award":["CG079801"],"award-info":[{"award-number":["CG079801"]}]},{"name":"NRF of Korea","award":["2018R1D1A1A02085584"],"award-info":[{"award-number":["2018R1D1A1A02085584"]}]},{"name":"NRF of Korea","award":["2021R1A2C1009639"],"award-info":[{"award-number":["2021R1A2C1009639"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2023,8]]},"abstract":"<jats:p> In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and [Formula: see text]-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure groups, pure fields and valued fields. More precisely, we prove Mekler\u2019s construction for groups, Chatzidakis\u2019 style criterion for pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras. <\/jats:p>","DOI":"10.1142\/s0219061322500210","type":"journal-article","created":{"date-parts":[[2022,6,17]],"date-time":"2022-06-17T09:18:28Z","timestamp":1655457508000},"source":"Crossref","is-referenced-by-count":8,"title":["On the antichain tree property"],"prefix":"10.1142","volume":"23","author":[{"given":"JinHoo","family":"Ahn","sequence":"first","affiliation":[{"name":"School of Computational Sciences, Korea Institute for Advanced Study (KIAS), 85 Hoegiro, Dongdaemun-gu, 02455 Seoul, South Korea"}]},{"given":"Joonhee","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, 03722 Seoul, South Korea"}]},{"given":"Junguk","family":"Lee","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Korea Advanced Institute of Science & Technology (KAIST), Daejeo, 291 Daehak-ro, Yuseong-gu, 34141 Daejeon, South Korea"}]}],"member":"219","published-online":{"date-parts":[[2022,12,17]]},"reference":[{"issue":"1","key":"S0219061322500210BIB003","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/S0168-0072(01)00088-4","volume":"115","author":"Baudisch A.","year":"2002","journal-title":"Ann. 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