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Earlier attempts to develop a fully nonflat semantics based on linear domain representations for a rich enough type system allowing inductive types, were designed in a way that felt rather artificial, as it featured certain awkward and counter\u2010intuitive properties; eventually, the focus turned on general, nonlinear representations. Here we try to turn this situation around, by showing that we can work linearly in a systematic way within the nonlinear model, and that we may even restrict to a fully linear model whose objects are in a bijective correspondence with the ones of the nonlinear and are easily seen to form a prime algebraic domain. To obtain our results we study mappings of finite approximations of objects that can be used to turn approximations into normal and linear forms.<\/jats:p>","DOI":"10.1002\/malq.201600020","type":"journal-article","created":{"date-parts":[[2018,4,17]],"date-time":"2018-04-17T04:26:24Z","timestamp":1523939184000},"page":"55-88","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Normal forms, linearity, and prime algebraicity over nonflat domains"],"prefix":"10.1002","volume":"64","author":[{"given":"Basil A.","family":"Kar\u00e1dais","sequence":"first","affiliation":[{"name":"Mathematisches Institut Ludwig\u2010Maximilians\u2010Universit\u00e4t Theresienstra\u00dfe 39 80333 M\u00fcnchen Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2018,4,16]]},"reference":[{"key":"e_1_2_9_2_1","first-page":"1","volume-title":"Handbook of Logic in Computer Science, Volume 3: Semantic Structures","author":"Abramsky S.","year":"1994"},{"key":"e_1_2_9_3_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511983504"},{"key":"e_1_2_9_4_1","volume-title":"Category Theory","author":"Awodey S.","year":"2010"},{"key":"e_1_2_9_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90038-F"},{"key":"e_1_2_9_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-08860-1_7"},{"key":"e_1_2_9_7_1","volume-title":"Cambridge Tracts in Theoretical Computer Science","author":"Bezem M.","year":"2003"},{"key":"e_1_2_9_8_1","doi-asserted-by":"publisher","DOI":"10.4204\/EPTCS.22.4"},{"key":"e_1_2_9_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.entcs.2007.02.011"},{"key":"e_1_2_9_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/152992.153008"},{"key":"e_1_2_9_11_1","volume-title":"Proofs and Types, Cambridge Tracts in Theoretical Computer Science","author":"Girard J. 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