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).
10.
M. Ferenczi, Partial transposition implies representability in cylindric algebras, Mathematical Logic Quarterly, 57 (1) (2011), 87-94.
11.Leon
Henkin, J.
Donald Monk, and Alfred
Tarski, Cylindric algebras. Part I, Studies in Logic and the
Foundations of Mathematics, vol. 64, North-Holland Publishing Co.,
Amsterdam, 1985. With an introductory chapter: General theory of algebras;
Reprint of the 1971 original. MR 781929
(86m:03095a)
13.Robin
Hirsch and Ian
Hodkinson, Relation algebras by games, Studies in Logic and
the Foundations of Mathematics, vol. 147, North-Holland Publishing
Co., Amsterdam, 2002. With a foreword by Wilfrid Hodges. MR 1935083
(2003m:03001)
15.Ildikó
Sain and Richard
J. Thompson, Strictly finite schema axiomatization of quasipolyadic
algebras, Algebraic logic (Budapest, 1988) Colloq. Math. Soc.
János Bolyai, vol. 54, North-Holland, Amsterdam, 1991,
pp. 539–571. MR 1153440
(93a:03072)
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)
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)
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)
M. Ferenczi, On the representability of neatly embeddable CA's by cylindric relativised algebras, Algebra Universalis, 63 (4) (2010), 331-350. MR 2734301
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)
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)
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)
Miklós Ferenczi Affiliation:
Department of Algebra, Budapest University of Technology and Economics, H–1521 Budapest, Hungary
Email:
ferenczi@math.bme.hu