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 is a subset of an -power . Beyond the usual set operations union, intersection, and complement, new operations are needed to describe such a field of sets, e.g., the th cylindrification the constant th diagonal the elementary substitution and the transposition for all restricted to the unit . 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

DOI:
http://dx.doi.org/10.1090/S0002-9947-2011-05332-8

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.