Cylindric algebra

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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[edit]

A cylindric algebra of dimension (where is any ordinal number) is an algebraic structure such that is a Boolean algebra, a unary operator on for every , and a distinguished element of for every and , such that the following hold:

(C1)

(C2)

(C3)

(C4)

(C5)

(C6) If , then

(C7) If , then

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

(C1)

(C2)

(C3)

(C4)

(C5)

(C6) If is a variable different from both and , then

(C7) If and are different variables, then

Generalizations[edit]

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[edit]

References[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.