{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T22:52:05Z","timestamp":1791586325706,"version":"4.3.4"},"reference-count":20,"publisher":"Duke University Press","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Notre Dame J. Formal Logic"],"published-print":{"date-parts":[[2010,1,1]]},"abstract":"<jats:p>We study completely decomposable torsion-free abelian groups of the form G S : = \u2295 n \u2208 S \u211a p n for sets S \u2286 \u03c9 . We show that G S has a decidable copy if and only if S is \u03a3 2 0 and has a computable copy if and only if S is \u03a3 3 0 .<\/jats:p>","DOI":"10.1215\/00294527-2010-006","type":"journal-article","created":{"date-parts":[[2010,5,4]],"date-time":"2010-05-04T16:39:47Z","timestamp":1272991187000},"page":"85-96","source":"Crossref","is-referenced-by-count":5,"title":["Decidability and Computability of Certain Torsion-Free Abelian Groups"],"prefix":"10.1215","volume":"51","author":[{"given":"Rodney G.","family":"Downey","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sergei S.","family":"Goncharov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Asher M.","family":"Kach","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Julia F.","family":"Knight","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Oleg V.","family":"Kudinov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alexander G.","family":"Melnikov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel","family":"Turetsky","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"73","reference":[{"key":"2","doi-asserted-by":"publisher","unstructured":"[2] Baer, R., \"Abelian groups without elements of finite order\", <i>Duke Mathematical Journal<\/i>, vol. 3 (1937), pp. 68--122.","DOI":"10.1215\/S0012-7094-37-00308-9"},{"key":"3","doi-asserted-by":"publisher","unstructured":"[3] Downey, R., and A. Montalb\u00e1n, \"The isomorphism problem for torsion-free abelian groups is analytic complete\", <i>Journal of Algebra<\/i>, vol. 320 (2008), pp. 2291--300.","DOI":"10.1016\/j.jalgebra.2008.06.007"},{"key":"4","unstructured":"[4] Ershov, Y. L., and S. S. Goncharov, <i>Constructive Models<\/i>, Siberian School of Algebra and Logic. Consultants Bureau, New York, 2000."},{"key":"5","unstructured":"[5] Ershov, Y. L., S. S. Goncharov, A. Nerode, J. B. Remmel, and V. W. Marek, editors, <i>Handbook of Recursive Mathematics. Vol. 2. Recursive Algebra, Analysis and Combinatorics<\/i>, vol. 139 of <i>Studies in Logic and the Foundations of Mathematics<\/i>, North-Holland, Amsterdam, 1998."},{"key":"6","doi-asserted-by":"publisher","unstructured":"[6] Feiner, L., \"The strong homogeneity conjecture\", <i>The Journal of Symbolic Logic<\/i>, vol. 35 (1970), pp. 375--77.","DOI":"10.2307\/2270693"},{"key":"7","doi-asserted-by":"publisher","unstructured":"[7] Feiner, L., \"Hierarchies of Boolean algebras\", <i>The Journal of Symbolic Logic<\/i>, vol. 35 (1970), pp. 365--74.","DOI":"10.2307\/2270692"},{"key":"8","doi-asserted-by":"publisher","unstructured":"[8] Fr\u00f6hlich, A., and J. C. Shepherdson, \"Effective procedures in field theory\", <i>Philosophical Transactions of the Royal Society of London. Series A<\/i>, vol. 248 (1956), pp. 407--32.","DOI":"10.1098\/rsta.1956.0003"},{"key":"9","unstructured":"[9] Fuchs, L., <i>Infinite Abelian Groups. Vol. II<\/i>, Academic Press, New York, 1973. Pure and Applied Mathematics. Vol. 36-II."},{"key":"10","unstructured":"[10] Griffor, E. R., editor, <i>Handbook of Computability Theory<\/i>, vol. 140 of <i>Studies in Logic and the Foundations of Mathematics<\/i>, North-Holland Publishing Co., Amsterdam, 1999."},{"key":"11","doi-asserted-by":"publisher","unstructured":"[11] Hjorth, G., \"The isomorphism relation on countable torsion free abelian groups\", <i>Fundamenta Mathematicae<\/i>, vol. 175 (2002), pp. 241--57.","DOI":"10.4064\/fm175-3-2"},{"key":"12","doi-asserted-by":"crossref","unstructured":"[12] Khisamiev, N. G., \"Constructive abelian groups\", pp. 1177--231 in <i>Handbook of Recursive Mathematics, Vol. 2<\/i>, vol. 139 of <i>Studies in Logic and the Foundations of Mathematics<\/i>, North-Holland, Amsterdam, 1998.","DOI":"10.1016\/S0049-237X(98)80050-5"},{"key":"13","doi-asserted-by":"crossref","unstructured":"[13] Khisamiev, N. G., \"On a class of strongly decomposable abelian groups\", <i>Algebra Logika<\/i>, vol. 41 (2002), pp. 493--509, 511--12.","DOI":"10.1023\/A:1020112806274"},{"key":"14","doi-asserted-by":"crossref","unstructured":"[14] Khisamiev, N. G., and A. A. Krykpaeva, \"Effectively completely decomposable abelian groups\", <i>Siberian Mathematical Journal<\/i>, vol. 38 (1997), pp. 1227--9.","DOI":"10.1007\/BF02675949"},{"key":"15","unstructured":"[15] Mal$'$cev, A. I., \"Constructive algebras. I\", <i>Uspekhi Matematicheskikh Nauk<\/i>, vol. 16 (1961), pp. 3--60."},{"key":"16","unstructured":"[16] Mal$'$cev, A. I., \"On recursive Abelian groups\", <i>Doklady Akademii Nauk SSSR<\/i>, vol. 146 (1962), pp. 1009--12."},{"key":"18","doi-asserted-by":"publisher","unstructured":"[18] Melnikov, A. G., \"Enumerations and completely decomposable torsion-free abelian groups\", <i>Theory of Computing Systems<\/i>, vol. 45 (2009), pp. 897--916.","DOI":"10.1007\/s00224-009-9175-9"},{"key":"19","doi-asserted-by":"publisher","unstructured":"[19] Metakides, G., and A. Nerode, \"Effective content of field theory\", <i>Annals of Mathematical Logic<\/i>, vol. 17 (1979), pp. 289--320.","DOI":"10.1016\/0003-4843(79)90011-1"},{"key":"20","doi-asserted-by":"publisher","unstructured":"[20] van der Waerden, B. L., \"Eine Bemerkung \u00fcber die Unzerlegbarkeit von Polynomen\", <i>Mathematische Annalen<\/i>, vol. 102 (1930), pp. 738--39.","DOI":"10.1007\/BF01782374"},{"key":"1","unstructured":"[1] Ash, C. J., and J. Knight, <i>Computable Structures and the Hyperarithmetical Hierarchy<\/i>, vol. 144 of <i>Studies in Logic and the Foundations of Mathematics<\/i>, North-Holland Publishing Co., Amsterdam, 2000."},{"key":"17","doi-asserted-by":"crossref","unstructured":"[17] Melnikov, A. G., \"$0^\\prime\\prime$-categorical completely decomposable torsion-free abelian groups\", pp. 362--71 in <i>Mathematical Theory and Computational Practice<\/i>, vol. 5635 of <i>Lecture Notes in Computer Science<\/i>, Springer, Berlin\/Heidelberg, 2009.","DOI":"10.1007\/978-3-642-03073-4_37"}],"container-title":["Notre Dame Journal of Formal Logic"],"original-title":[],"link":[{"URL":"https:\/\/projecteuclid.org\/journalArticle\/Download?urlid=10.1215\/00294527-2010-006","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T22:22:30Z","timestamp":1791584550000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/notre-dame-journal-of-formal-logic\/volume-51\/issue-1\/Decidability-and-Computability-of-Certain-Torsion-Free-Abelian-Groups\/10.1215\/00294527-2010-006.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1,1]]},"references-count":20,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,1]]}},"URL":"https:\/\/doi.org\/10.1215\/00294527-2010-006","relation":{},"ISSN":["0029-4527","1939-0726"],"issn-type":[{"value":"0029-4527","type":"print"},{"value":"1939-0726","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1,1]]}}}