## The polyadic generalization of the Boolean axiomatization of fields of sets

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**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).## References

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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