Bodirsky, Manuel; Macpherson, Dugald Reducts of structures and maximal-closed permutation groups. (English) Zbl 1372.03060 J. Symb. Log. 81, No. 3, 1087-1114 (2016). The paper concerns the question by M. Junker and M. Ziegler [ibid. 73, No. 3, 861–884 (2008; Zbl 1189.03041)] whether any non-\(\omega\)-categorical structure has infinitely many proper non-trivial reducts. The authors give a negative answer to this question and construct a non-\(\omega\)-categorical structure which has no proper non-trivial reducts in both the model-theoretic and the group-theoretic sense. This example has a D-relation and its study is connected with Jordan permutation groups. They also construct a non-\(\omega\)-categorical strongly minimal set which has no proper non-trivial reducts in the model-theoretic sense. Reviewer: Beibut Kulpeshov (Almaty) Cited in 4 Documents MSC: 03C35 Categoricity and completeness of theories 03C40 Interpolation, preservation, definability 03C60 Model-theoretic algebra 03C10 Quantifier elimination, model completeness, and related topics 20B35 Subgroups of symmetric groups 20B27 Infinite automorphism groups Keywords:reducts; D-relations; Jordan permutation groups; maximal closed subgroups; strongly minimal sets Citations:Zbl 1189.03041 PDFBibTeX XMLCite \textit{M. Bodirsky} and \textit{D. Macpherson}, J. Symb. Log. 81, No. 3, 1087--1114 (2016; Zbl 1372.03060) Full Text: DOI arXiv References: [1] Uncountably Categorical Theories (1993) [2] DOI: 10.1016/0168-0072(95)00037-2 · Zbl 0858.03039 [3] A Course in Model Theory 40 (2012) [4] Journal of the London Mathematical Society 42 pp 64– (1990) [5] Automorphisms of First Order Structures (1994) · Zbl 0797.00010 [6] DOI: 10.1007/BF02579282 · Zbl 0492.05036 [7] Oligomorphic Permutation Groups 152 (1990) · Zbl 0813.20002 [8] Classification Theory (Proceedings, Chicago, 1985) 1292 pp 132– (1987) [9] DOI: 10.1007/BF01214702 · Zbl 0313.20022 [10] Model Theory (1993) [11] European Journal of Mathematics 11 pp 17– (2013) [12] Algebra i Logika 29 pp 368– (1990) [13] DOI: 10.1090/conm/558/11058 [14] Introduction to {\(\Lambda\)}-Trees (2001) · Zbl 1004.20014 [15] Relations Related to Betweenness: Their Structure and Automorphisms 131 (1998) · Zbl 0896.08001 [16] Proceedings of the London Mathematical Society 72 pp 63– (1996) [17] Classification Theory and the Number of Non-isomorphic Models (1990) [18] Oxford Logic Guides 32 (1996) [19] DOI: 10.1016/j.aim.2014.08.008 · Zbl 1403.03053 [20] Proceedings of the London Mathematical Society 50 pp 265– (1985) [21] DOI: 10.1017/S0963548304006716 · Zbl 1059.05103 [22] DOI: 10.1112/blms/12.4.303 · Zbl 0443.20001 [23] DOI: 10.1016/0168-0072(95)00061-5 · Zbl 0865.03025 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.