{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T03:16:29Z","timestamp":1770520589628,"version":"3.49.0"},"reference-count":20,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2022,8,20]],"date-time":"2022-08-20T00:00:00Z","timestamp":1660953600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2022,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o\u2010minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good\u2010shaped definable subsets called quasi\u2010special submanifolds. Using this fact, we investigate definably complete locally o\u2010minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o\u2010minimal expansions of ordered fields, \u0141ojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo\u2010definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo\u2010definable spaces are definably compact. Here, a pseudo\u2010definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set\u2010valued functions with definably compact domains of\u00a0definition.<\/jats:p>","DOI":"10.1002\/malq.202200004","type":"journal-article","created":{"date-parts":[[2022,8,20]],"date-time":"2022-08-20T14:56:42Z","timestamp":1661007402000},"page":"496-515","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Tameness of definably complete locally o\u2010minimal structures and definable bounded multiplication"],"prefix":"10.1002","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1289-5182","authenticated-orcid":false,"given":"Masato","family":"Fujita","sequence":"first","affiliation":[{"name":"Department of Liberal Arts Japan Coast Guard Academy 5\u20101 Wakaba\u2010cho, Kure Hiroshima 737\u20108512 Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tomohiro","family":"Kawakami","sequence":"additional","affiliation":[{"name":"Department of Mathematics Wakayama University Wakayama 640\u20108510 Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wataru","family":"Komine","sequence":"additional","affiliation":[{"name":"Institute of Mathematics University of Tsukuba Ibaraki 305\u20108571 Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2022,8,20]]},"reference":[{"key":"e_1_2_7_2_1","volume-title":"Set Valued Analysis","author":"Aubin J.\u2010B.","year":"1990"},{"key":"e_1_2_7_3_1","volume-title":"Folge \/ A Series of Modern Surveys in Mathematics","author":"Bochnak J.","year":"1998"},{"key":"e_1_2_7_4_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511525919"},{"key":"e_1_2_7_5_1","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-96-08416-1"},{"key":"e_1_2_7_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2012.10.002"},{"key":"e_1_2_7_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2019.102756"},{"key":"e_1_2_7_8_1","doi-asserted-by":"publisher","DOI":"10.1017\/jsl.2020.31"},{"key":"e_1_2_7_9_1","doi-asserted-by":"crossref","unstructured":"M.Fujita Locally o\u2010minimal structures with tame topological properties J. Symb. Log. DOI: 10.1017\/jsl.2021.80 to appear.","DOI":"10.1017\/jsl.2021.80"},{"key":"e_1_2_7_10_1","doi-asserted-by":"crossref","unstructured":"M.Fujita Functions definable in definably complete uniformly locally o\u2010minimal structure of the second kind arXiv:2010.02420 (2020).","DOI":"10.1017\/jsl.2020.31"},{"key":"e_1_2_7_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2022.103144"},{"key":"e_1_2_7_12_1","volume-title":"Fun With Fields","author":"Johnson W.","year":"2016"},{"key":"e_1_2_7_13_1","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/06430783"},{"key":"e_1_2_7_14_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1994-1195484-5"},{"key":"e_1_2_7_15_1","doi-asserted-by":"publisher","DOI":"10.2307\/2694974"},{"key":"e_1_2_7_16_1","volume-title":"Grundlehren der mathematischen Wissenschaften","author":"Rockafeller R. 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