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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The polyadic generalization of the Boolean axiomatization of fields of sets
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by Miklós Ferenczi PDF
Trans. Amer. Math. Soc. 364 (2012), 867-886 Request permission

Abstract:

A version of the classical representation theorem for Boolean algebras states that the fields of sets form a variety and that a possible axiomatization is the system of Boolean axioms. An important case for fields of sets occurs when the unit $V$ is a subset of an $\alpha$-power $^{\alpha }U$. Beyond the usual set operations union, intersection, and complement, new operations are needed to describe such a field of sets, e.g., the $i$th cylindrification $C_{i},$ the constant $ij$th diagonal $D_{ij},$ the elementary substitution $\left [ i\;/\;j\right ]$ and the transposition $\left [ i,\;j\right ]$ for all $i,j<\alpha$ restricted to the unit $V$. Here it is proven that such generalized fields of sets being closed under the above operations form a variety; further, a first order finite scheme axiomatization of this variety is presented. In the proof a crucial role is played by the existence of the operator transposition. The foregoing axiomatization is close to that of finitary polyadic equality algebras (or quasi-polyadic equality algebras).
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Additional Information
  • Miklós Ferenczi
  • Affiliation: Department of Algebra, Budapest University of Technology and Economics, H–1521 Budapest, Hungary
  • Email: ferenczi@math.bme.hu
  • Received by editor(s): September 4, 2009
  • Received by editor(s) in revised form: February 23, 2010
  • Published electronically: September 8, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 867-886
  • MSC (2010): Primary 03G15, 03G05, 03C05; Secondary 03G25, 03C48, 03C95
  • DOI: https://doi.org/10.1090/S0002-9947-2011-05332-8
  • MathSciNet review: 2846356