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The polyadic generalization of the Boolean axiomatization of fields of sets

Author: Miklós Ferenczi
Journal: Trans. Amer. Math. Soc. 364 (2012), 867-886
MSC (2010): Primary 03G15, 03G05, 03C05; Secondary 03G25, 03C48, 03C95
Published electronically: September 8, 2011
MathSciNet review: 2846356
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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

Keywords: Representation, Stone theorem, polyadic algebras, fields of sets, algebraic logic, cylindric algebras.
Received by editor(s): September 4, 2009
Received by editor(s) in revised form: February 23, 2010
Published electronically: September 8, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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