Positive Jonsson theories. (English. French summary) Zbl 1456.03060
Summary: This paper is a general introduction to Positive Logic, where only what we call h-inductive sentences are under consideration, allowing the extension to homomorphisms of model-theoric notions which are classically associated to embeddings; in particular, the existentially closed models, that were primitively defined by Abraham Robinson, become here positively closed models. It accounts for recent results in this domain, and is oriented towards the positivisation of Jonsson theories.
MSC:
| 03C95 | Abstract model theory |
| 03C07 | Basic properties of first-order languages and structures |
| 03C10 | Quantifier elimination, model completeness, and related topics |
| 03C52 | Properties of classes of models |
References:
| [1] | Barwise, J.: Handbook of Mathematical Logic, Model Theory, vol. 1. North Holland, Amsterdam (1982) · Zbl 0623.03002 |
| [2] | Ben Yaacov, I, Positive model theory and compact abstract theories, J. Math. Log., 3, 85-118, (2003) · Zbl 1028.03034 · doi:10.1142/S0219061303000212 |
| [3] | Ben Yaacov, I, Thickness, and a categoric point of view of type-space functors, Fundam. Math., 179, 199-224, (2003) · Zbl 1044.03023 · doi:10.4064/fm179-3-2 |
| [4] | Ben Yaacov, I, Simplicity in compact abstract theories, J. Math. Log., 3, 163-191, (2004) · Zbl 1039.03032 · doi:10.1142/S0219061303000297 |
| [5] | Ben Yaacov, I, Lovely pairs of models: the non first order case, J. Symb. Log., 69, 641-662, (2004) · Zbl 1073.03017 · doi:10.2178/jsl/1096901759 |
| [6] | Ben Yaacov, I, Uncountable dense categoricity in cats, J. Symb. Log., 70, 829-860, (2005) · Zbl 1084.03028 · doi:10.2178/jsl/1122038916 |
| [7] | Ben Yaacov, I; Poizat, B, Fondements de la logique positive, J. Symb. Log., 72, 1141-1162, (2007) · Zbl 1139.03022 · doi:10.2178/jsl/1203350777 |
| [8] | Belkasmi, M, Positive model theory and amalgamation, Notre Dame J. Form. Log., 55, 205-230, (2014) · Zbl 1351.03025 · doi:10.1215/00294527-2420648 |
| [9] | Cheng-Chung, C., Keisler, J.H.: Model Theory. North-Holland, Amsterdam (1973) · Zbl 0276.02032 |
| [10] | Daigneault, A.: Théorie des Modèles en Logique Mathématique, Université de Montréeal (1967) · Zbl 0164.31001 |
| [11] | Fraïssé, R.: Sur l’extension aux relations de quelques propriétés connues des ordres. C.R. Acad. Sci. Paris 237, 508-510 (1953) · Zbl 0053.02904 |
| [12] | Gödel, K, Die vollständigkeit der axiome des logischen funktionenkalkuls, Monat. Math. Phys., 37, 349-360, (1930) · JFM 56.0046.04 · doi:10.1007/BF01696781 |
| [13] | Jonsson, B, Universal relational systems, Math. Scand., 4, 193-208, (1956) · Zbl 0077.25302 · doi:10.7146/math.scand.a-10468 |
| [14] | Jonsson, B, Homogeneous universal relational systems, Math. Scand., 8, 137-142, (1960) · Zbl 0173.00505 · doi:10.7146/math.scand.a-10601 |
| [15] | Kaiser, K, Über eine verallgemeinerung der robinsonschen modell-vervollständigung, Z. Math. Logik Grundlagen Math., 15, 37-48, (1969) · Zbl 0188.32101 · doi:10.1002/malq.19690150107 |
| [16] | Kunghozhin, A, Existentially closed and maximal models in positive logic (en russe), Algebra Log., 54, 496-506, (2013) · Zbl 1295.03020 · doi:10.1007/s10469-013-9209-x |
| [17] | Makkai, M., Reyes, G.: First Order Categorical Logic. Lectures Notes in Mathematics, vol. 611. Springer, Berlin (1977) · Zbl 0357.18002 |
| [18] | Makowski, JH, On some conjectures connected with complete sentences, Fundam. Math., 81, 193-202, (1974) · Zbl 0285.02042 · doi:10.4064/fm-81-3-193-202 |
| [19] | Morley, M; Vaught, R, Homogeneous universal models, Math. Scand., 11, 37-57, (1963) · Zbl 0112.00603 · doi:10.7146/math.scand.a-10648 |
| [20] | Mustafin, T, Jonsson generalized conditions and generalized Jonsson theories of Boolean algebras (in Russian), math. trudy, Novosibirsk, 135-197; translated, Siberian Advances in Mathematics, 10, 1-58, (1998) · Zbl 0913.03038 |
| [21] | Mustafin, Y, Quelques propriétés des théories de Jonsson, J. Symb. Log., 67, 528-536, (2002) · Zbl 1013.03046 · doi:10.2178/jsl/1190150095 |
| [22] | Mustafin, T., Nurkhaïdarov, E: Jonsson theories of polygons over a group (in Russian), Sbornik nauchnyh trudov, Qaraghandy, 67-73 (1995) |
| [23] | Nurkhaïdarov, E.: Jonsson theories of abelian groups (in Russian), Sbornik nauchnyh trudov, Qaraghandy, 43-50 (1995) · Zbl 0077.25302 |
| [24] | Nurtazin, A.T..: Countable existentially closed models of universally axiomatizable theories (in Russian), SORAN Trudy po Matematike, xx, 1-50 (2015) · Zbl 1374.03025 |
| [25] | Pillay, A.: Forking in the category of existentially closed structures, Quaderni di Matematica, vol. 6 (2000) · Zbl 0986.03031 |
| [26] | Poizat, B, Univers positifs, J. Symb. Log., 71, 969-976, (2006) · Zbl 1110.03022 · doi:10.2178/jsl/1154698585 |
| [27] | Poizat, B, Quelques effets pervers de la positivité, Ann. Pure Appl. Log., 6, 812-816, (2010) · Zbl 1226.03047 · doi:10.1016/j.apal.2009.06.012 |
| [28] | Poizat, B.: Parlons ladakhi. L’Harmattan (2018) · Zbl 1073.03017 |
| [29] | Robinson, A.: Complete Theories. North Holland, Amsterdam (1956) · Zbl 0070.02701 |
| [30] | Yeshkeyev, A.: Description of Jonsson theories of unars (in Russian), Sbornik nauchnyh trudov, Qaraghandy, 51-57 (1995) · Zbl 1028.03034 |
| [31] | Yeshkeyev, A.: Jonsson theories (in Russian), Qaraghandy (2009) · Zbl 1456.03060 |
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.
