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The regular algebra of a poset


Author: Pere Ara
Journal: Trans. Amer. Math. Soc. 362 (2010), 1505-1546
MSC (2000): Primary 16D70; Secondary 16E50, 06F05, 46L80
DOI: https://doi.org/10.1090/S0002-9947-09-04884-3
Published electronically: October 20, 2009
MathSciNet review: 2563739
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Abstract: Let $ K$ be a fixed field. We attach to each finite poset $ \mathbb{P}$ a von Neumann regular $ K$-algebra $ Q_K(\mathbb{P})$ in a functorial way. We show that the monoid of isomorphism classes of finitely generated projective $ Q_K(\mathbb{P})$-modules is the abelian monoid generated by $ \mathbb{P}$ with the only relations given by $ p=p+q$ whenever $ q<p$ in $ \mathbb{P}$. This extends the class of monoids for which there is a positive solution to the realization problem for von Neumann regular rings.


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Additional Information

Pere Ara
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain
Email: para@mat.uab.cat

DOI: https://doi.org/10.1090/S0002-9947-09-04884-3
Keywords: von Neumann regular ring, poset, primitive monoid, Toeplitz algebra, Leavitt path algebra
Received by editor(s): February 15, 2008
Published electronically: October 20, 2009
Additional Notes: This research was partially supported by the DGI and European Regional Development Fund, jointly, through Project MTM2005-00934, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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