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Euler characteristics for strongly minimal groups and the EQ-expansions of vector spaces. (English) Zbl 1220.03023

Summary: 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.

MSC:

03C60 Model-theoretic algebra
03C50 Models with special properties (saturated, rigid, etc.)
03C65 Models of other mathematical theories
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References:

[1] Model theory 42 (1993)
[2] Invariant measures on groups satisfying various chain conditions 76 pp 209– (2011)
[3] Division rings whose vector spaces are pseudofinite 75 pp 1087– (2010)
[4] Handbook of proof theory 137 (1998) · Zbl 0898.03001
[5] DOI: 10.1090/S0002-9939-1993-1116261-6 · doi:10.1090/S0002-9939-1993-1116261-6
[6] DOI: 10.1002/malq.19750210145 · Zbl 0312.02045 · doi:10.1002/malq.19750210145
[7] DOI: 10.1090/S0002-9947-07-04382-6 · Zbl 1127.03026 · doi:10.1090/S0002-9947-07-04382-6
[8] Algebra 211 (2002) · Zbl 0984.00001
[9] DOI: 10.2307/421058 · Zbl 0968.03036 · doi:10.2307/421058
[10] Geometric stability theory 32 (1996)
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