Cylindric algebra

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The notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This is comparable to the role Boolean algebras play for propositional logic. Indeed, cylindric algebras are Boolean algebras equipped with additional cylindrification operations that model quantification and equality. They differ from polyadic algebras in that the latter do not model equality.

Definition of a cylindric algebra[edit]

A cylindric algebra of dimension \alpha (where \alpha is any ordinal number) is an algebraic structure (A,+,\cdot,-,0,1,c_\kappa,d_{\kappa\lambda})_{\kappa,\lambda<\alpha} such that (A,+,\cdot,-,0,1) is a Boolean algebra, c_\kappa a unary operator on A for every \kappa, and d_{\kappa\lambda} a distinguished element of A for every \kappa and \lambda, such that the following hold:

(C1) c_\kappa 0=0

(C2) x\leq c_\kappa x

(C3) c_\kappa(x\cdot c_\kappa y)=c_\kappa x\cdot c_\kappa y

(C4) c_\kappa c_\lambda x=c_\lambda c_\kappa x

(C5) d_{\kappa\kappa}=1

(C6) If \kappa\notin\{\lambda,\mu\}, then d_{\lambda\mu}=c_\kappa(d_{\lambda\kappa}\cdot d_{\kappa\mu})

(C7) If \kappa\neq\lambda, then c_\kappa(d_{\kappa\lambda}\cdot x)\cdot c_\kappa(d_{\kappa\lambda}\cdot -x)=0

Assuming a presentation of first-order logic without function symbols, the operator c_\kappa x models existential quantification over variable \kappa in formula x while the operator d_{\kappa\lambda} models the equality of variables \kappa and \lambda. Henceforth, reformulated using standard logical notations, the axioms read as

(C1) \exists \kappa. \mathit{false} \Leftrightarrow \mathit{false}

(C2) x \Rightarrow \exists \kappa. x

(C3) \exists \kappa. (x\wedge \exists \kappa. y) \Leftrightarrow (\exists\kappa. x) \wedge (\exists\kappa. y)

(C4) \exists\kappa \exists\lambda. x \Leftrightarrow \exists \lambda \exists\kappa. x

(C5) \kappa=\kappa \Leftrightarrow \mathit{true}

(C6) If \kappa is a variable different from both \lambda and \mu, then \lambda=\mu \Leftrightarrow \exists\kappa. (\lambda=\kappa \wedge \kappa=\mu)

(C7) If \kappa and \lambda are different variables, then \exists\kappa. (\kappa=\lambda \wedge x) \wedge \exists\kappa. (\kappa=\lambda\wedge \neg x) \Leftrightarrow \mathit{false}


Recently, cylindric algebras have been generalized to the many-sorted case, which allows for a better modeling of the duality between first-order formulas and terms.

See also[edit]


Further reading[edit]

  • Imieliński, T.; Lipski, W. (1984). "The relational model of data and cylindric algebras". Journal of Computer and System Sciences 28: 80–102. doi:10.1016/0022-0000(84)90077-1.