Cylindric algebra

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

A cylindric algebra of dimension $\alpha$, where $\alpha$ is any ordinal 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 $\kappa\neq\lambda\mu$, 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) $\exists \kappa \bot = \bot$

(C2) $x \leq \exists \kappa x$

(C3) $\exists \kappa (x\wedge \exists \kappa y) = \exists\kappa x\wedge \exists\kappa y$

(C4) $\exists\kappa \exists\lambda x = \exists \lambda \exists\kappa x$

(C5) $d_{\kappa\kappa} = \top$

(C6) If $\kappa\neq\lambda\mu$, then $d_{\lambda\mu}=\exists\kappa(d_{\lambda\kappa}\wedge d_{\kappa\mu})$

(C7) If $\kappa\neq\lambda$, then $\exists\kappa(d_{\kappa\lambda}\wedge x)\wedge \exists\kappa(d_{\kappa\lambda}\wedge \neg x)=\bot$

Generalizations

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.