. Exemple, Les corps algébriquement clos (resp. réel clos) sont les modèles existentiellement clos de la théorie des corps (resp. des corps ordonnés) dans le langage des anneaux L ann, resp. des anneaux ordonnés L ann ? { })

, Une extension élémentaire de N est une extension M de N qui satisfait les mêmes formules (à paramètres dans N )

. Exemple and . Soit-t-une-théorie-quelconque, Supposons que la classe ? des modèles existentiellement clos de T soit élémentaire. Alors on peut montrer que la théorie de ? est modèle-complète

, Une théorie T * est le modèle-compagnon d'une théorie T si les modèles de T * sont exactement les modèles existentiellement clos de T . Si de plus T a la propriété d'amalgation

. Exemple, La théorie des corps algébriquement clos (resp. réel clos) est la modèle-complétion

, Une théorie T élimine les quantificateurs si toute formule du langage de T est logiquement équivalente modulo T à une formule sans quanteur

, Un théorie T universelle admet une modèle-complétion T * si et seulement si T * est un modèle-compagnon de T et si T * élimine les quantificateurs

;. Astier and V. Astier, Elementary equivalence of lattices of open sets definable in ominimal expansions of real closed fields, Fund. Math, vol.220, issue.1, pp.7-21, 2013.

L. Bélair, Panorama of p-adic model theory, Ann. Sci. Math. Québec, vol.36, issue.1, pp.43-75, 2012.

F. Bellissima, Finitely generated free Heyting algebras, J. Symbolic Logic, vol.51, issue.1, pp.152-165, 1986.

[. Bochnak, Real algebraic geometry, volume 36 de Ergebnisse der Mathematik und ihrer Grenzgebiete, de Graduate Texts in Mathematics, vol.90, 1983.

R. Cluckers, Classification of semi-algebraic p-adic sets up to semialgebraic bijection, J. Reine Angew. Math, vol.540, pp.105-114, 2001.

R. Cluckers, Presburger sets and P -minimal fields, J. Symbolic Logic, vol.68, issue.1, pp.153-162, 2003.

R. Cluckers, Analytic p-adic cell decomposition and integrals, Trans. Amer. Math. Soc, vol.356, issue.4, pp.1489-1499, 2004.

. Cluckers, Local metric properties and regular stratifications of p-adic definable sets, Comment. Math. Helv, vol.87, issue.4, pp.963-1009, 2012.

R. Cluckers and E. Leenknegt, A version of p-adic minimality, J. Symbolic Logic, vol.77, issue.2, pp.621-630, 2012.

M. ;. Cluckers, R. Cluckers, and F. Martin, A definable p-adic analogue of Kirszbraun's theorem on extensions of Lipschitz maps, J. Inst. Math. Jussieu, vol.17, issue.1, pp.39-57, 2018.

. Cubides-kovacsics, Topological cell decomposition and dimension theory in p-minimal fields, J. Symb. Log, vol.82, issue.1, pp.347-358, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01188341

. Cubides-kovacsics, ;. Leenknegt, P. Cubides-kovacsics, and E. Leenknegt, Integration and cell decomposition in P-minimal structures, J. Symb. Log, vol.81, issue.3, pp.1124-1141, 2016.

P. Cubides-kovacsics and K. H. Nguyen, A P -minimal structure without definable Skolem functions, J. Symb. Log, vol.82, issue.2, pp.778-786, 2017.

L. Darnière, Model-completion of scaled lattices. Prépublications mathématiques d'Angers 191, Département de mathématiques de l'université d, Angers, 2004.

;. Darnière, L. Darnière, and L. Darnière, Polytopes and simplexes in p-adic fields, On the model-completion of heyting algebras, vol.168, pp.1284-1307, 2017.

. Darnière and L. Darnière, Model completion of scaled lattices and co-heyting algebras of p-adic semi-algebraic sets, Math. Logic Quart, 2019.

. Darnière and L. Darnière, Semi-algebraic triangulation over p-adically closed fields, Proc. Lond. Math. Soc, vol.118, issue.6, pp.1501-1546, 2019.

L. Darnière and I. Halpuczok, Cell decomposition and classification of definable sets in p-optimal fields, J. Symb. Log, vol.82, issue.1, pp.120-136, 2017.

. Darnière, L. Darnière, and M. Junker, On Bellissima's construction of the finitely generated free Heyting algebras, and beyond, Arch. Math. Logic, vol.49, issue.7-8, pp.743-771, 2010.

. Darnière, L. Darnière, and M. Junker, Codimension and pseudometric in co-Heyting algebras, Algebra Universalis, vol.64, issue.3, pp.251-282, 2011.

. Darnière, L. Darnière, and M. Junker, Model completion of varieties of co-Heyting algebras, Houston J. Math, vol.44, issue.1, pp.49-82, 2018.

L. Darnière and M. Tressl, Defining integer valued functions in rings of continuous definable functions over a topological field, J. Math. Log, 2019.

;. Denef and J. Denef, The rationality of the Poincaré series associated to the p-adic points on a variety, Invent. Math, vol.77, issue.1, pp.1-23, 1984.

J. Denef, p-adic semi-algebraic sets and cell decomposition, J. Reine Angew. Math, vol.369, pp.154-166, 1986.

J. Denef, . Van-den, and L. Dries, p-adic and real subanalytic sets, Ann. of Math, vol.128, issue.2, pp.79-138, 1988.

A. Fehm and F. Jahnke, Recent progress on definability of Henselian valuations, Ordered algebraic structures and related topics, vol.697, pp.135-143, 2017.

;. Et-zawadowski, S. Ghilardi, and M. Zawadowski, Model completions and r-Heyting categories, Ann. Pure Appl. Logic, vol.88, issue.1, pp.27-46, 1997.

[. Green, On Rumely's local-global principle, Jahresber. Deutsch. Math.-Verein, vol.97, issue.2, pp.43-74, 1995.

A. Grzegorczyk, Undecidability of some topological theories, Fund. Math, vol.38, pp.137-152, 1951.

I. Halupczok, Non-Archimedean Whitney stratifications, Proc. Lond. Math. Soc, vol.109, issue.3, pp.1304-1362, 2014.

. Haskell, . Macpherson, D. Haskell, and D. Macpherson, Cell decompositions of Cminimal structures, Ann. Pure Appl. Logic, vol.66, issue.2, pp.113-162, 1994.

. Haskell, . Macpherson, D. Haskell, and D. Macpherson, A version of o-minimality for the p-adics, J. Symbolic Logic, vol.62, issue.4, pp.1075-1092, 1997.

M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc, vol.142, pp.43-60, 1969.

T. Hosoi, On intermediate logics, I. J. Fac. Sci. Univ. Tokyo Sect. I, vol.14, pp.293-312, 1967.

[. Jahnke, Dp-minimal valued fields, J. Symb. Log, vol.82, issue.1, pp.151-165, 2017.

;. Macintyre and A. Macintyre, On definable subsets of p-adic fields, J. Symbolic Logic, vol.41, issue.3, pp.605-610, 1976.

;. Mathews and L. Mathews, Cell decomposition and dimension functions in first-order topological structures, Proc. London Math. Soc, vol.70, issue.3, pp.1-32, 1995.

;. Moret-bailly and L. Moret-bailly, Groupes de Picard et problèmes de Skolem. I, II, Ann. Sci. École Norm. Sup, vol.22, issue.4, pp.181-194, 1989.

M. Mourgues, Corps p-minimaux avec fonctions de skolem définissables, 1999.

M. Mourgues, Cell decomposition for P -minimal fields, MLQ Math. Log. Q, vol.55, issue.5, pp.487-492, 2009.

;. Pitts and A. M. Pitts, On an interpretation of second-order quantification in first-order intuitionistic propositional logic, J. Symbolic Logic, vol.57, issue.1, pp.33-52, 1992.

A. Prestel and P. Roquette, Formally p-adic fields, Lecture Notes in Math, vol.1050, 1984.

A. Prestel and J. Schmid, Existentially closed domains with radical relations, J. Reine Angew. Math, vol.407, pp.178-201, 1990.

D. Richard, Definability in terms of the successor function and the coprimeness predicate in the set of arbitrary integers, J. Symbolic Logic, vol.54, issue.4, pp.1253-1287, 1989.

R. S. Rumely, Arithmetic over the ring of all algebraic integers, J. Reine Angew. Math, vol.368, pp.127-133, 1986.

P. Scowcroft, . Van-den, and L. Dries, On the structure of semialgebraic sets over p-adic fields, J. Symbolic Logic, vol.53, issue.4, pp.1138-1164, 1988.

;. Simon and P. Simon, On dp-minimal ordered structures, J. Symbolic Logic, vol.76, issue.2, pp.448-460, 2011.

. Simon, P. Simon, and E. Walsberg, Tame topology over dp-minimal structures, Notre Dame J. of Formal Logic, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01207708

M. Tressl, Elimination theory for the ring of algebraic integers, Contemp. Math, vol.388, pp.189-205, 1988.

D. Van-den and L. , Dimension of definable sets, algebraic boundedness and Henselian fields, Stability in model theory, vol.45, pp.189-209, 1987.

D. Van-den and L. , Tame topology and o-minimal structures, vol.248, 1998.