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Journal articles on the topic 'Strongly minimal'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Rao, K. Chandrasekhara, and P. Padma P . Padma. "Strongly Minimal Generalized Boundary." Indian Journal of Applied Research 1, no. 7 (October 1, 2011): 176–77. http://dx.doi.org/10.15373/2249555x/apr2012/59.

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2

Hrushovski, Ehud, and James Loveys. "Strongly and co-strongly minimal abelian structures." Journal of Symbolic Logic 75, no. 2 (June 2010): 442–58. http://dx.doi.org/10.2178/jsl/1268917489.

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AbstractWe give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);2. when the theory of the structure is strongly minimal.In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d ∈ D. the index of A ∩ dA, in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.

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3

Eleftheriou, Pantelis E., Assaf Hasson, and Ya'acov Peterzil. "Strongly minimal groups in o-minimal structures." Journal of the European Mathematical Society 23, no. 10 (May 28, 2021): 3351–418. http://dx.doi.org/10.4171/jems/1095.

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4

Hillman, Jonathan A. "Strongly minimal PD4-complexes." Topology and its Applications 156, no. 8 (April 2009): 1565–77. http://dx.doi.org/10.1016/j.topol.2009.01.006.

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5

Hasson, Assaf, and Ehud Hrushovski. "DMP in strongly minimal sets." Journal of Symbolic Logic 72, no. 3 (September 2007): 1019–30. http://dx.doi.org/10.2178/jsl/1191333853.

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AbstractWe construct a strongly minimal set which is not a finite cover of one with DMP. We also show that for a strongly minimal theory T, generic automorphisms exist iff T has DMP, thus proving a conjecture of Kikyo and Pillay.

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6

Blossier, Thomas, and Elisabeth Bouscaren. "Finitely axiomatizable strongly minimal groups." Journal of Symbolic Logic 75, no. 1 (March 2012): 25–50. http://dx.doi.org/10.2178/jsl/1264433908.

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7

Hrushovski, Ehud. "A new strongly minimal set." Annals of Pure and Applied Logic 62, no. 2 (July 1993): 147–66. http://dx.doi.org/10.1016/0168-0072(93)90171-9.

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8

Hong, Chan Yong, and Tai Keun Kwak. "On minimal strongly prime ideals." Communications in Algebra 28, no. 10 (January 2000): 4867–78. http://dx.doi.org/10.1080/00927870008827127.

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9

Sun, Weihua, Yuming Xu, and Ning Li. "On minimal strongly KC-spaces." Czechoslovak Mathematical Journal 59, no. 2 (June 2009): 305–16. http://dx.doi.org/10.1007/s10587-009-0022-6.

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10

Ravi, O., S. Jeyashri, and K. Vijayalakshmi. "On Strongly Minimal \(g\)-Continuous Functions in Minimal Structures." Journal of Advanced Studies in Topology 1, no. 2 (December 1, 2010): 29. http://dx.doi.org/10.20454/jast.2010.212.

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11

Andrews, Uri, and Julia Knight. "Strongly minimal theories with recursive models." Journal of the European Mathematical Society 20, no. 7 (May 15, 2018): 1561–94. http://dx.doi.org/10.4171/jems/793.

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12

Chisholm, J., J. F. Knight, and S. Miller. "Computable embeddings and strongly minimal theories." Journal of Symbolic Logic 72, no. 3 (September 2007): 1031–40. http://dx.doi.org/10.2178/jsl/1191333854.

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AbstractHere we prove that if T and T′ are strongly minimal theories, where T′ satisfies a certain property related to triviality and T does not, and T′ is model complete, then there is no computable embedding of Mod(T) into Mod(T′). Using this, we answer a question from [4], showing that there is no computable embedding of VS into ZS, where VS is the class of infinite vector spaces over ℚ, and ZS is the class of models of Th(ℤ, S). Similarly, we show that there is no computable embedding of ACF into ZS, where ACF is the class of algebraically closed fields of characteristic 0.

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13

Andrews, Uri, Steffen Lempp, and Noah Schweber. "Building models of strongly minimal theories." Advances in Mathematics 386 (August 2021): 107802. http://dx.doi.org/10.1016/j.aim.2021.107802.

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14

L'Innocente, Sonia, Vera Puninskaya, and Carlo Toffalori. "Strongly Minimal Modules Over Group Rings." Communications in Algebra 33, no. 7 (June 2005): 2089–107. http://dx.doi.org/10.1081/agb-200063504.

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15

Hillman, Jonathan A. "PD4-complexes with strongly minimal models." Topology and its Applications 153, no. 14 (August 2006): 2413–24. http://dx.doi.org/10.1016/j.topol.2005.09.002.

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16

Holland, Kitty L. "Strongly minimal fusions of vector spaces." Annals of Pure and Applied Logic 83, no. 1 (January 1997): 1–22. http://dx.doi.org/10.1016/0168-0072(95)00006-2.

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17

KOWALSKI, PIOTR, and SERGE RANDRIAMBOLOLONA. "STRONGLY MINIMAL REDUCTS OF VALUED FIELDS." Journal of Symbolic Logic 81, no. 2 (June 2016): 510–23. http://dx.doi.org/10.1017/jsl.2015.61.

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AbstractWe prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.

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18

Zil'ber, B. I. "Strongly minimal countably categorical theories. III." Siberian Mathematical Journal 25, no. 4 (1985): 559–71. http://dx.doi.org/10.1007/bf00968893.

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19

Zil'ber, B. I. "Strongly minimal countably categorical theories. II." Siberian Mathematical Journal 25, no. 3 (1985): 396–412. http://dx.doi.org/10.1007/bf00968979.

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20

Baronti, M., and G. Lewicki. "Strongly Unique Minimal Projections on Hyperplanes." Journal of Approximation Theory 78, no. 1 (July 1994): 1–18. http://dx.doi.org/10.1006/jath.1994.1056.

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21

Ikeda, Koichiro. "Minimal but not strongly minimal structures with arbitrary finite dimensions." Journal of Symbolic Logic 66, no. 1 (March 2001): 117–26. http://dx.doi.org/10.2307/2694913.

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AbstractAn infinite structure is said to be minimal if each of its definable subset is finite or cofinite. Modifying Hrushovski's method we construct minimal, non strongly minimal structures with arbitrary finite dimensions. This answers negatively to a problem posed by B. I Zilber.

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22

Hasson, Assaf, and Piotr Kowalski. "Strongly minimal expansions of (ℂ, +) definable in o-minimal fields." Proceedings of the London Mathematical Society 97, no. 1 (January 4, 2008): 117–54. http://dx.doi.org/10.1112/plms/pdm052.

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23

Blossier, Thomas. "Automorphism groups of trivial strongly minimal structures." Journal of Symbolic Logic 68, no. 2 (June 2003): 644–68. http://dx.doi.org/10.2178/jsl/1052669069.

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AbstractWe study automorphism groups of trivial strongly minimal structures. First we give a characterization of structures of bounded valency through their groups of automorphisms. Then we characterize the triplets of groups which can be realized as the automorphism group of a non algebraic component, the subgroup stabilizer of a point and the subgroup of strong automorphisms in a trivial strongly minimal structure, and also we give a reconstruction result. Finally, using HNN extensions we show that any profinite group can be realized as the stabilizer of a point in a strongly minimal structure of bounded valency.

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24

Marker, David. "Non Σn axiomatizable almost strongly minimal theories." Journal of Symbolic Logic 54, no. 3 (September 1989): 921–27. http://dx.doi.org/10.2307/2274752.

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Recall that a theory is said to be almost strongly minimal if in every model every element is in the algebraic closure of a strongly minimal set. In 1970 Hodges and Macintyre conjectured that there is a natural number n such that every ℵ0-categorical almost strongly minimal theory is Σn axiomatizable. Recently Ahlbrandt and Baldwin [A-B] proved that if T is ℵ0-categorical and almost strongly minimal, then T is Σn axiomatizable for some n. This result also follows from Ahlbrandt and Ziegler's results on quasifinite axiomatizability [A-Z]. In this paper we will refute Hodges and Macintyre's conjecture by showing that for each n there is an ℵ0-categorical almost strongly minimal theory which is not Σn axiomatizable.Before we begin we should note that in all these examples the complexity of the theory arises from the complexity of the definition of the strongly minimal set. It is still open whether the conjecture is true if we allow a predicate symbol for the strongly minimal set.We will prove the following result.Theorem. For every n there is an almost strongly minimal ℵ0-categorical theory T with models M and N such that N is Σn elementary but not Σn + 1 elementary.To show that these theories yield counterexamples to the conjecture we apply the following result of Chang [C].Theorem. If T is a Σn axiomatizable theory categorical in some infinite power, M and N are models of T and N is a Σn elementary extension of M, then N is an elementary extension of M.

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25

Buechler, Steven. "Pseudoprojective strongly minimal sets are locally projective." Journal of Symbolic Logic 56, no. 4 (December 1991): 1184–94. http://dx.doi.org/10.2307/2275467.

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AbstractLet D be a strongly minimal set in the language L, and D′ ⊃ D an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T′ be the theory of the structure (D′, D), where D interprets the predicate D. It is known that T′ is ω-stable. We proveTheorem A. If D is not locally modular, then T′ has Morley rank ω.We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a, b ∈ D and closed X ⊂ D, a ∈ cl(Xb) ⇒ there is a Y ⊂ X with a ∈ cl(Yb) and ∣Y∣ ≤ k. Using Theorem A, we proveTheorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective.The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B.Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular.

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26

Marker, David. "A strongly minimal expansion of (ω, s)." Journal of Symbolic Logic 52, no. 1 (March 1987): 205–7. http://dx.doi.org/10.2307/2273874.

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We will show that there is a nontrivial strongly minimal expansion of (ω, s), the natural numbers with successor. Pillay and Steinhorn [1] proved that there is no -minimal expansion of (ω, ≤). This result provides an interesting contrast.The strongly minimal expansion of (ω, s) is very easy to describe. Consider the order-two permutation of ω, π recursively defined byLet T be Th(ω, s, π, 0).

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27

Puninskaya, V. A. "Strongly minimal modules over right distributive rings." Algebra and Logic 35, no. 3 (May 1996): 196–203. http://dx.doi.org/10.1007/bf02367218.

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28

Hrushovski, Ehud. "Strongly minimal expansions of algebraically closed fields." Israel Journal of Mathematics 79, no. 2-3 (October 1992): 129–51. http://dx.doi.org/10.1007/bf02808211.

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29

Lee, Heakyung. "Strongly right FBN rings." Bulletin of the Australian Mathematical Society 38, no. 3 (December 1988): 457–64. http://dx.doi.org/10.1017/s0004972700027842.

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The annihilator of a finite generated β-critical module is called a β-coprimative ideal. A prime ideal P is called β-prime if the Krull dimension of R/P is β. This paper is concerned with the relationship between the set of β-prime ideals and the set of minimal β-coprimitive ideals over a strongly right FBN ring. it is shown that there exists a one-to-one correspondence between the set of β-prime ideals and the set on minimal β-coprimitive ideals over a strongly right FBN ring R for −1 < β ≤ α, where α is the Krull dimension of R.

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30

Marker, David. "Omitting types in -minimal theories." Journal of Symbolic Logic 51, no. 1 (March 1986): 63–74. http://dx.doi.org/10.2307/2273943.

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Let L be a first order language containing a binary relation symbol <.Definition. Suppose ℳ is an L-structure and < is a total ordering of the domain of ℳ. ℳ is ordered minimal (-minimal) if and only if any parametrically definable X ⊆ ℳ can be represented as a finite union of points and intervals with endpoints in ℳ.In any ordered structure every finite union of points and intervals is definable. Thus the -minimal structures are the ones with no unnecessary definable sets. If T is a complete L-theory we say that T is strongly (-minimal if and only if every model of T is -minimal.The theory of real closed fields is the canonical example of a strongly -minimal theory. Strongly -minimal theories were introduced (in a less general guise which we discuss in §6) by van den Dries in [1]. Extending van den Dries' work, Pillay and Steinhorn (see [3], [4] and [2]) developed an extensive structure theory for definable sets in strongly -minimal theories, generalizing the results for real closed fields. They also established several striking analogies between strongly -minimal theories and ω-stable theories (most notably the existence and uniqueness of prime models). In this paper we will examine the construction of models of strongly -minimal theories emphasizing the problems involved in realizing and omitting types. Among other things we will prove that the Hanf number for omitting types for a strongly -minimal theory T is at most (2∣T∣)+, and characterize the strongly -minimal theories with models order isomorphic to (R, <).

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31

Chudnovsky, Maria, Cemil Dibek, and Paul Seymour. "New examples of minimal non-strongly-perfect graphs." Discrete Mathematics 344, no. 5 (May 2021): 112334. http://dx.doi.org/10.1016/j.disc.2021.112334.

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32

Holland, Kitty L. "Model completeness of the new strongly minimal sets." Journal of Symbolic Logic 64, no. 3 (September 1999): 946–62. http://dx.doi.org/10.2307/2586613.

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Boris Zil'ber conjectured that all strongly minimal theories are bi-interpretable with one of the “classical” sorts: theories of algebraically closed fields, theories of infinite vector spaces over division rings and theories with trivial algebraic closure relations. Hrushovski produced the first two classes of counterexample to this conjecture in [10] and [9]. Subsequently, in [8], the author gave an explicit axiomatization of a special case of [9] from which model completeness could quickly be deduced. It was unclear at that writing whether the model completeness result was true in the general case or was due to peculiarities of the case under consideration. The main new result of this paper is model completeness, not only of the general case in [9], but also of the theories described in [10]. Specifically, we present a general framework in which producing a strongly minimal theory is reduced to finding an elementary class of theories satisfying certain requirements (see below). We present the theories of [10] and [9] as special instances of such theories, giving an explicit axiomatization from which model completeness immediately follows in each case.We hope by presenting these constructions in parallel, using common language and extracting common elements, to make easier both the exploitation of the ideas involved in their making and their comparison with other recent constructions of a similar flavor. For a selection of such constructions, see [6], [1], [2] and [3]. For more general background, see [2], [4] and [11].

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33

Moosa, Rahim, and Anand Pillay. "$ℵ_{0}$-categorical strongly minimal compact complex manifolds." Proceedings of the American Mathematical Society 140, no. 5 (May 1, 2012): 1785–801. http://dx.doi.org/10.1090/s0002-9939-2011-11028-1.

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34

Baldwin, John T. "An almost strongly minimal non-Desarguesian projective plane." Transactions of the American Mathematical Society 342, no. 2 (February 1, 1994): 695–711. http://dx.doi.org/10.1090/s0002-9947-1994-1165085-8.

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35

Serrano, Alberto Castellón, Antonio Fernández López, Amable García Martín, and Cándido Martín González. "Strongly prime alternative pairs with minimal inner ideals." Manuscripta Mathematica 90, no. 1 (December 1996): 479–87. http://dx.doi.org/10.1007/bf02568320.

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36

Lewicki, Grzegorz, and Agnieszka Micek. "Equality of two strongly unique minimal projection constants." Journal of Approximation Theory 162, no. 12 (December 2010): 2278–89. http://dx.doi.org/10.1016/j.jat.2010.07.014.

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37

Khachiyan, Leonid, Endre Boros, Khaled Elbassioni, and Vladimir Gurvich. "On Enumerating Minimal Dicuts and Strongly Connected Subgraphs." Algorithmica 50, no. 1 (October 27, 2007): 159–72. http://dx.doi.org/10.1007/s00453-007-9074-x.

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38

HYTTINEN, TAPANI. "INTERPRETING GROUPS INSIDE MODULAR STRONGLY MINIMAL HOMOGENEOUS MODELS." Journal of Mathematical Logic 03, no. 01 (May 2003): 127–42. http://dx.doi.org/10.1142/s0219061303000200.

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A large homogeneous (not necessarily saturated) model M is strongly minimal, if any definable subset is either bounded or has bounded complement. In this case (M, bcl) is a pregeometry, where bcl denotes the bounded closure operation. In this paper, we show that if M is a large homogeneous strongly minimal structure and (M, bcl) is non-trivial and locally modular, then M interprets a group. In addition, we give a description of such groups.

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39

Harbrecht, Helmut, Wolfgang L. Wendland, and Natalia Zorii. "Minimal energy problems for strongly singular Riesz kernels." Mathematische Nachrichten 291, no. 1 (July 31, 2017): 55–85. http://dx.doi.org/10.1002/mana.201600024.

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40

Ziegler, Martin. "An exposition of Hrushovskiʼs New Strongly Minimal Set." Annals of Pure and Applied Logic 164, no. 12 (December 2013): 1507–19. http://dx.doi.org/10.1016/j.apal.2013.06.020.

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41

Xue, Weimin. "Structure of minimal noncommutative duo rings and minimal strongly bounded non-duo rings." Communications in Algebra 20, no. 9 (January 1992): 2777–88. http://dx.doi.org/10.1080/00927879208824488.

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42

Şahi̇n, Mesut, and Leah Gold Stella. "Gluing semigroups and strongly indispensable free resolutions." International Journal of Algebra and Computation 29, no. 02 (March 2019): 263–78. http://dx.doi.org/10.1142/s0218196719500012.

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We study strong indispensability of minimal free resolutions of semigroup rings focusing on the operation of gluing used in the literature to take examples with a special property and produce new ones. We give a naive condition to determine whether gluing of two semigroup rings has a strongly indispensable minimal free resolution. As applications, we determine simple gluings of [Formula: see text]-generated non-symmetric, [Formula: see text]-generated symmetric and pseudo symmetric numerical semigroups as well as obtain infinitely many new complete intersection semigroups of any embedding dimensions, having strongly indispensable minimal free resolutions.

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43

Kalimoldayev, M. N., A. T. Nurtazin, and Z. G. Khisamiyev. "TWO PROPERTIES OF EXISTENTIALLY CLOSED COMPANIONS OF STRONGLY MINIMAL STRUCTURES." PHYSICO-MATHEMATICAL SERIES 5, no. 333 (October 15, 2020): 28–32. http://dx.doi.org/10.32014/2020.2518-1726.79.

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The proposed article studies some properties of existentially closed companions of strongly minimal structures. A criterion for the existential closedness of an arbitrary strongly minimal structure is found in the article and it is proved that the existentially closed companion of any strongly minimal structure is itself strongly minimal. It also follows from the resulting description that all existentially closed companions of a given strongly minimal structure form an axiomatizable class whose elementary theory is complete and model-complete and, therefore, coincides with its inductive and forcing companions. This is the reason for the importance of the work done and the high international significance of the theorems obtained in it. Another equally important consequence of this research is the discovery of an important subclass of strongly minimal theories. It should be noted that a complete description of this class of theories is an independent and extremely important task. It is known that natural numbers with the following relation are an example of a strongly minimal structure in which the existential type of zero is not minimal. Then the method used in the proof of the last theorem shows that the existentially closed companion of this structure are integers with the following relation.

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44

Dalud-Vincent, Monique. "Minimal closed subsets and strongly connected components in pretopology." Applied Mathematical Sciences 15, no. 6 (2021): 265–81. http://dx.doi.org/10.12988/ams.2021.914485.

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45

Kulpeshov, B. Sh, and S. V. Sudoplatov. "Properties of ranks for families of strongly minimal theories." Sibirskie Elektronnye Matematicheskie Izvestiya 19, no. 1 (February 14, 2022): 120–24. http://dx.doi.org/10.33048/semi.2022.19.011.

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46

Mirzayanov, M. R. "On Minimal Strongly Connected Congruences of a Directed Path." Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics 6, no. 1/2 (2006): 91–95. http://dx.doi.org/10.18500/1816-9791-2006-6-1-2-91-95.

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47

Andrews, U. "Spectra of recursive models of disintegrated strongly minimal theories." Lobachevskii Journal of Mathematics 35, no. 4 (October 2014): 287–91. http://dx.doi.org/10.1134/s1995080214040118.

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48

Buechler, Steven. "One theorem of Zil′ber's on strongly minimal sets." Journal of Symbolic Logic 50, no. 4 (December 1985): 1054–61. http://dx.doi.org/10.2307/2273990.

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AbstractSuppose D ⊂ M is a strongly minimal set definable in M with parameters from C. We say D is locally modular if for all X, Y ⊂ D, with X = acl(X ∪ C)∩D, Y = acl(Y ∪ C) ∩ D and X ∩ Y ≠ ∅,We prove the following theorems.Theorem 1. Suppose M is stable and D ⊂ M is strongly minimal. If D is not locally modular then inMeqthere is a definable pseudoplane.(For a discussion of Meq see [M, §A].) This is the main part of Theorem 1 of [Z2] and the trichotomy theorem of [Z3].Theorem 2. Suppose M is stable and D, D′ ⊂ M are strongly minimal and nonorthogonal. Then D is locally modular if and only if D′ is locally modular.

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49

Giudice, Gian F., and Alexander Kusenko. "A strongly-interacting phase of the minimal supersymmetric model." Physics Letters B 439, no. 1-2 (October 1998): 55–62. http://dx.doi.org/10.1016/s0370-2693(98)00996-4.

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50

Subha, E., and N. Nagaveni. "Strongly Minimal Generalized Closed Set in Biminimal Structure Spaces." Procedia Computer Science 47 (2015): 394–99. http://dx.doi.org/10.1016/j.procs.2015.03.222.

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