The notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of 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.

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

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.

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
• First-order logic
• Relation algebras (RA)
• Polyadic equality algebras (PEA)

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.
• Caleiro, C., and Gonçalves, R (2007) "On the algebraization of many-sorted logics" in J. Fiadeiro and P.-Y. Schobbens, eds., Recent Trends in Algebraic Development Techniques - Selected Papers, Vol. 4409 of Lecture Notes in Computer Science. Springer-Verlag: 21-36.

External links

• Jipsen's algebra page.

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