Journal of Symbolic Logic

Euler characteristics for strongly minimal groups and the eq-expansions of vector spaces

Vinicius Cifú Lopes

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Abstract

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 complete one to the eq-expansion of those spaces while preserving the Fubini property but not completeness. Then, surjective interpretable endofunctions in those spaces are injective, and conversely. Our presentation is made in the general setting of multi-sorted structures.

Article information

Source
J. Symbolic Logic, Volume 76, Issue 1 (2011), 235-242.

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1294170998

Digital Object Identifier
doi:10.2178/jsl/1294170998

Mathematical Reviews number (MathSciNet)
MR2791346

Zentralblatt MATH identifier
1220.03023

Citation

Cifú Lopes, Vinicius. Euler characteristics for strongly minimal groups and the eq-expansions of vector spaces. J. Symbolic Logic 76 (2011), no. 1, 235--242. doi:10.2178/jsl/1294170998. https://projecteuclid.org/euclid.jsl/1294170998


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