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  1. (1 other version)Euler characteristics for strongly minimal groups and the eq-expansions of vector spaces.Vinicius Cifú Lopes - 2011 - Journal of Symbolic Logic 76 (1):235 - 242.
    We find the complete Euler characteristics for the categories of definable sets and functions in strongly minimal groups. Their images, which represent the Grothendieck semirings of those categories, are all isomorphic to the semiring of polynomials over the integers with nonnegative leading coefficient. As a consequence, injective definable endofunctions in those groups are surjective. For infinite vector spaces over arbitrary division rings, the same results hold, and more: We also establish the Fubini property for all Euler characteristics, and extend the (...)
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  • Approximate Euler characteristic, dimension, and weak pigeonhole principles.Jan Krajíček - 2004 - Journal of Symbolic Logic 69 (1):201-214.
    We define the notion of approximate Euler characteristic of definable sets of a first order structure. We show that a structure admits a non-trivial approximate Euler characteristic if it satisfies weak pigeonhole principle WPHP2nn: two disjoint copies of a non-empty definable set A cannot be definably embedded into A, and principle CC of comparing cardinalities: for any two definable sets A, B either A definably embeds in B or vice versa. Also, a structure admitting a non-trivial approximate Euler characteristic must (...)
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  • (1 other version)Euler characteristics for strongly minimal groups and the eq-expansions of vector spaces.Vinicius Cifú Lopes - 2011 - Journal of Symbolic Logic 76 (1):235-242.
    We find the complete Euler characteristics for the categories of definable sets and functions in strongly minimal groups. Their images, which represent the Grothendieck semirings of those categories, are all isomorphic to the semiring of polynomials over the integers with nonnegative leading coefficient. As a consequence, injective definable endofunctions in those groups are surjective. For infinite vector spaces over arbitrary division rings, the same results hold, and more: We also establish the Fubini property for all Euler characteristics, and extend the (...)
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  • Invariant measures in simple and in small theories.Artem Chernikov, Ehud Hrushovski, Alex Kruckman, Krzysztof Krupiński, Slavko Moconja, Anand Pillay & Nicholas Ramsey - 2023 - Journal of Mathematical Logic 23 (2).
    We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism invariant Keisler measure [Formula: see text] and (ii) a definable group [Formula: see text] in a simple theory such that [Formula: see text] is not definably amenable, i.e. there is no translation invariant Keisler measure on [Formula: see text]. We also discuss paradoxical decompositions both in the setting of discrete groups (...)
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  • Multidimensional Exact Classes, Smooth Approximation and Bounded 4-Types.Daniel Wolf - 2020 - Journal of Symbolic Logic 85 (4):1305-1341.
    In connection with the work of Anscombe, Macpherson, Steinhorn and the present author in [1] we investigate the notion of a multidimensional exact class (R-mec), a special kind of multidimensional asymptotic class (R-mac) with measuring functions that yield the exact sizes of definable sets, not just approximations. We use results about smooth approximation [24] and Lie coordinatization [13] to prove the following result (Theorem 4.6.4), as conjectured by Macpherson: For any countable language$\mathcal {L}$and any positive integerdthe class$\mathcal {C}(\mathcal {L},d)$of all (...)
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  • A remark on divisibility of definable groups.Mário J. Edmundo - 2005 - Mathematical Logic Quarterly 51 (6):639-641.
    We show that if G is a definably compact, definably connected definable group defined in an arbitrary o-minimal structure, then G is divisible. Furthermore, if G is defined in an o-minimal expansion of a field, k ∈ ℕ and pk : G → G is the definable map given by pk = xk for all x ∈ G , then we have |–1| ≥ kr for all x ∈ G , where r > 0 is the maximal dimension of abelian (...)
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  • Grothendieck Ring of the Pairing Function without Cycles.Esther Elbaz - 2022 - Notre Dame Journal of Formal Logic 63 (2).
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  • O-minimalism.Hans Schoutens - 2014 - Journal of Symbolic Logic 79 (2):355-409.
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  • Integration in algebraically closed valued fields with sections.Yimu Yin - 2013 - Annals of Pure and Applied Logic 164 (1):1-29.
    We construct Hrushovski–Kazhdan style motivic integration in certain expansions of ACVF. Such an expansion is typically obtained by adding a full section or a cross-section from the RV-sort into the VF-sort and some extra structure in the RV-sort. The construction of integration, that is, the inverse of the lifting map , is rather straightforward. What is a bit surprising is that the kernel of is still generated by one element, exactly as in the case of integration in ACVF. The overall (...)
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  • Weakly o-minimal nonvaluational structures.Roman Wencel - 2008 - Annals of Pure and Applied Logic 154 (3):139-162.
    A weakly o-minimal structure image expanding an ordered group is called nonvaluational iff for every cut left angle bracketC,Dright-pointing angle bracket of definable in image, we have that inf{y−x:xset membership, variantC,yset membership, variantD}=0. The study of nonvaluational weakly o-minimal expansions of real closed fields carried out in [D. Macpherson, D. Marker, C. Steinhorn,Weakly o-minimal structures and real closed fields, Trans. Amer. Math. Soc. 352 5435–5483. MR1781273 (2001i:03079] suggests that this class is very close to the class of o-minimal expansions of (...)
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  • Definable combinatorics with dense linear orders.Himanshu Shukla, Arihant Jain & Amit Kuber - 2020 - Archive for Mathematical Logic 59 (5-6):679-701.
    We compute the model-theoretic Grothendieck ring, \\), of a dense linear order with or without end points, \\), as a structure of the signature \, and show that it is a quotient of the polynomial ring over \ generated by \\) by an ideal that encodes multiplicative relations of pairs of generators. This ring can be embedded in the polynomial ring over \ generated by \. As a corollary we obtain that a DLO satisfies the pigeon hole principle for definable (...)
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  • Geometric properties of semilinear and semibounded sets.Jana Maříková - 2006 - Mathematical Logic Quarterly 52 (2):190-202.
    We calculate the universal Euler characteristic and universal dimension function on semilinear and semibounded sets and obtain some criteria for definable equivalence of semilinear and semibounded sets in terms of these invariants.
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  • Grothendieck rings of theories of modules.Amit Kuber - 2015 - Annals of Pure and Applied Logic 166 (3):369-407.
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  • On the Euler characteristic of definable groups.Mário J. Edmundo - 2011 - Mathematical Logic Quarterly 57 (1):44-46.
    We show that in an arbitrary o-minimal structure the following are equivalent: conjugates of a definable subgroup of a definably connected, definably compact definable group cover the group if the o-minimal Euler characteristic of the quotient is non zero; every infinite, definably connected, definably compact definable group has a non trivial torsion point.
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