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

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

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).

**1.**H. Andréka, S. D. Comer, J. X. Madarász, I. Németi and T. Sayed Ahmed,*Epimorphisms in cylindric algebras and definability in finite variable logic*, Algebra Universalis,**61**(3-4) (2009), 261-282. MR**2565854 (2011a:03070)****2.**H. Andréka and R. J. Thompson,*A Stone type representation theorem for algebras of relations of higher rank*, Transaction of Amer. Math. Soc.,**309**(2) (1988), 671-682. MR**961607 (90d:03135)****3.**H. Andréka,*A finite axiomatization of locally square cylindric-relativized set algebras*, Studia Sci. Math. Hun.,**38**(1-4) (2001), 1-11. MR**1877766 (2002m:03097)****4.**H. Andréka, M. Ferenczi and I. Németi,*Cylindric-like Algebras and Algebraic Logic*, Bolyai Society Mathematical Studies, Springer, to appear.**5.**S. D. Comer,*Galois theory of cylindric algebras and its applications*, Transaction of Amer. Math. Soc.,**286**(1984), 771-785. MR**760986 (85m:03044)****6.**A. Daigneault and J. D. Monk,*Representation theory for polyadic algebras*, Fund. Math.,**52**(1963), 151-176. MR**0151373 (27:1358)****7.**M. Ferenczi,*On the representability of neatly embeddable CA's by cylindric relativised algebras*, Algebra Universalis,**63**(4) (2010), 331-350. MR**2734301****8.**M. Ferenczi and G. Sági,*On some developments in the representation theory of cylindric-like algebras*, Algebra Universalis,**55**(2-3) (2006), 345-353. MR**2280236 (2007j:03095)****9.**M. Ferenczi,*On cylindric algebras satisfying merry-go-round properties*, Logic Journal of IGPL,**15**(2) (2007), 183-197. MR**2333809 (2008i:03073)****10.**M. Ferenczi,*Partial transposition implies representability in cylindric algebras*, Mathematical Logic Quarterly,**57**(1) (2011), 87-94.**11.**L. Henkin, J. D. Monk and A. Tarski,*Cylindric Algebras I-II*, North Holland, 1985. MR**0781929 (86m:03095a); MR0781930 (86m:03095b)****12.**R. Hirsch and I. Hodkinson,*Step by step building representations in algebraic logic*, J. Symbolic Logic,**62**(1) (1997), 225-279. MR**1450522 (98g:03145)****13.**R. Hirsch and I. Hodkinson,*Relation Algebras by Games*, North Holland, 2002. MR**1935083 (2003m:03001)****14.**J. D. Monk,*Nonfinitazibility of classes of representable cylindric algebras*, J. Symbolic Logic,**34**(1969), 331-343. MR**0256861 (41:1517)****15.**I. Sain and R. J. Thompson,*Strictly finite schema axiomatization of quasi-polyadic algebras*, in Algebraic Logic, Coll. Math. Soc. J. Bolyai, 1988, pp. 539-571. MR**1153440 (93a:03072)****16.**Tarek Sayed Ahmed,*On complete representations of reducts of polyadic algebras*, Studia Logica,**89**(3) (2008), 325-332. MR**2438547 (2009g:03103)**

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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:
https://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.