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Mat. Zametki, 2007, Volume 81, Issue 6, Pages 842–854 (Mi mz3735)  

Reducibility of Monadic Equivalence Relations

V. G. Kanoveia, V. A. Lyubetskiia, M. Reekenb

a Institute for Information Transmission Problems, Russian Academy of Sciences
b University of Wuppertal

Abstract: Each additive cut in the nonstandard natural numbers $ ^*{\mathbb N}$ induces the equivalence relation $\operatorname M_U$ on $ ^*{\mathbb N}$ defined as $x\operatorname M_Uy$ if $|x-y|\in U$. Such equivalence relations are said to be monadic. Reducibility between monadic equivalence relations is studied. The main result (Theorem 3.1) is that reducibility can be defined in terms of cofinality (or coinitiality) and a special parameter of a cut, called its width. Smoothness and the existence of transversals are also considered. The results obtained are similar to theorems of modern descriptive set theory on the reducibility of Borel equivalence relations.

Keywords: nonstandard analysis, additive cut of the hyperintegers, monadic equivalence relation, $\kappa$-determined set, $\kappa$-determined reducibility, width of a cut

DOI: https://doi.org/10.4213/mzm3735

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English version:
Mathematical Notes, 2007, 81:6, 757–766

Bibliographic databases:

UDC: 510.2
Received: 20.12.2005
Revised: 24.08.2006

Citation: V. G. Kanovei, V. A. Lyubetskii, M. Reeken, “Reducibility of Monadic Equivalence Relations”, Mat. Zametki, 81:6 (2007), 842–854; Math. Notes, 81:6 (2007), 757–766

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