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.

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

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