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Tame and wild refinement monoids

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Abstract

The class of refinement monoids (commutative monoids satisfying the Riesz refinement property) is subdivided into those which are tame, defined as being an inductive limit of finitely generated refinement monoids, and those which are wild, i.e., not tame. It is shown that tame refinement monoids enjoy many positive properties, including separative cancellation (\(2x=2y=x+y \implies x=y\)) and multiplicative cancellation with respect to the algebraic ordering (\(mx\le my \implies x\le y\)). In contrast, examples are constructed to exhibit refinement monoids which enjoy all the mentioned good properties but are nonetheless wild.

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Acknowledgments

We thank the referee for his/her thorough reading of the manuscript and for useful suggestions, particularly Proposition 3.16, and we thank E. Pardo and F. Wehrung for helpful correspondence and references. Part of this research was undertaken while the second-named author held a sabbatical fellowship from the Ministerio de Educación y Ciencias de España at the Centre de Recerca Matemàtica in Barcelona during spring 2011. He thanks both institutions for their support and hospitality. The first-named author was partially supported by DGI MINECO MTM2011-28992-C02-01, by FEDER UNAB10-4E-378 “Una manera de hacer Europa”, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

    P. Ara

  2. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    K. R. Goodearl

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  1. P. Ara
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Correspondence to K. R. Goodearl.

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Communicated by László Márki.

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Ara, P., Goodearl, K.R. Tame and wild refinement monoids. Semigroup Forum 91, 1–27 (2015). https://doi.org/10.1007/s00233-014-9647-3

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  • DOI: https://doi.org/10.1007/s00233-014-9647-3

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