Atom Structures of Cylindric Algebras and Relation Algebras

For any finite n 3 there are two atomic n-dimensional cylindric algebras with the same atom structure, with one representable, the other, not. Hence, the complex algebra of the atom structure of a representable atomic cylindric algebra is not always representable, so that the class RCAn of representable n-dimensional cylindric algebras is not closed under completions. Further, it follows by an argument of Venema that RCAn is not axiomatisable by Sahlqvist equations, and hence nor by equations where negation can only occur in constant terms. Similar results hold for relation algebras.

[1]  Ian M. Hodkinson,et al.  Step by step – Building representations in algebraic logic , 1997, Journal of Symbolic Logic.

[2]  Roger D. Maddux,et al.  Finite integral relation algebras , 1985 .

[3]  K. Jon Barwise,et al.  On Moschovakis closure ordinals , 1977, Journal of Symbolic Logic.

[4]  D. Monk.  On representable relation algebras. , 1964 .

[5]  Ian M. Hodkinson,et al.  Complete representations in algebraic logic , 1997, Journal of Symbolic Logic.

[6]  Johan van Benthem,et al.  Modal Languages and Bounded Fragments of Predicate Logic , 1998, J. Philos. Log..

[7]  Y. Venema.  Atom structures and Sahlqvist equations , 1997 .

[8]  M. Stone.  The theory of representations for Boolean algebras , 1936 .

[9]  A. Tarski,et al.  Cylindric Algebras. Part II , 1988 .

[10]  A. Tarski,et al.  Boolean Algebras with Operators. Part I , 1951 .

[11]  J. P. Andersen.  The Representation of Relation Algebras , 1961 .

[12]  C. J. Everett,et al.  The Representation of Relational Algebras. , 1951 .

[13]  Robin Hirsch,et al.  Representability is not decidable for finite relation algebras , 1999 .

[14]  R. Maddux.  Some varieties containing relation algebras , 1982 .

[15]  Hajnal Andréka,et al.  Complexity of Equations Valid in Algebras of Relations: Part II: Finite Axiomatizations , 1997, Ann. Pure Appl. Log..

[16]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[17]  J. Donald Monk,et al.  Nonfinitizability of Classes of Representable Cylindric Algebras , 1969, J. Symb. Log..

[18]  A. Tarski,et al.  Boolean Algebras with Operators , 1952 .

[19]  R. Goldblatt.  Elementary generation and canonicity for varieties of Boolean algebras with operators , 1995 .

[20]  J. Donald Monk,et al.  Completions of BOOLEAN Algebras with operators , 1970 .

[21]  András Simon,et al.  Notions of Density That Imply Representability in Algebraic Logic , 1998, Ann. Pure Appl. Log..

[22]  J. Benthem.  Some kinds of modal completeness , 1980 .

[23]  Hajnal Andréka,et al.  Complexity of Equations Valid in Algebras of Relations: Part I: Strong Non-Finitizability , 1997, Ann. Pure Appl. Log..