Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Natural numerosities of sets of tuples


Authors: Marco Forti and Giuseppe Morana Roccasalvo
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 03E65, 03F25; Secondary 03A05, 03C20
Published electronically: July 2, 2014
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a notion of ``numerosity'' for sets of tuples of natural numbers that satisfies the five common notions of Euclid's Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a notion, we show that, contrasting to cardinal arithmetic, the natural ``Cantorian'' definitions of order relation and arithmetical operations provide a very good algebraic structure. In fact, numerosities can be taken as the non-negative part of a discretely ordered ring, namely the quotient of a formal power series ring modulo a suitable (``gauge'') ideal. In particular, special numerosities, called ``natural'', can be identified with the semiring of hypernatural numbers of appropriate ultrapowers of $ \mathbb{N}$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 03E65, 03F25, 03A05, 03C20

Retrieve articles in all journals with MSC (2010): 03E65, 03F25, 03A05, 03C20


Additional Information

Marco Forti
Affiliation: Dipartimento di Matematica, University of Pisa, Via Buonarroti 1C, 56100 Pisa, Italy
Email: forti@dma.unipi.it

Giuseppe Morana Roccasalvo
Affiliation: Dipartimento di Matematica, University of Pisa, Via Buonarroti 1C, 56100 Pisa, Italy
Email: moranaroccasalvo@mail.dm.unipi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06136-9
PII: S 0002-9947(2014)06136-9
Received by editor(s): June 18, 2012
Received by editor(s) in revised form: December 2, 2012
Published electronically: July 2, 2014
Additional Notes: The first author’s research was partially supported by MIUR Grant PRIN 2009, Italy.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.