Cylindric algebra

The notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of equational first-order logic. 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

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 $\exists\kappa. (\kappa=\lambda \wedge x) \wedge \exists\kappa. (\kappa=\lambda\wedge \neg x) \Leftrightarrow \mathit{false}$

Generalizations

Cylindric algebras have been generalized to the case of many-sorted logic (Caleiro and Gonçalves 2006), which allows for a better modeling of the duality between first-order formulas and terms.

See also

• Abstract algebraic logic
• Lambda calculus and Combinatory logic, other approaches to modelling quantification and eliminating variables
• Hyperdoctrines are a categorical formulation of cylindric algebras
• Relation algebras (RA)
• Polyadic algebra

References

• Leon Henkin, Monk, J.D., and Alfred Tarski (1971) Cylindric Algebras, Part I. North-Holland. ISBN 978-0-7204-2043-2.
• -------- (1985) Cylindric Algebras, Part II. North-Holland.
• Carlos Caleiro, Ricardo Gonçalves (2006). "On the algebraization of many-sorted logics" (PDF). In J. Fiadeiro and P.-Y. Schobbens. Proc. 18th int. conf. on Recent trends in algebraic development techniques (WADT). LNCS 4409. Springer. pp. 21–36. ISBN 978-3-540-71997-7.

Further reading

• 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.