In mathematics, the notion of cylindric algebra, developed 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.
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.
The cylindric algebra should not be confused with the measure theoretic concept cylindrical algebra that arises in the study of cylinder set measures and the cylindrical σ-algebra.
A cylindric algebra of dimension
is any ordinal number) is an algebraic structure
κ λ
κ , λ < α
κ λ
(called a diagonal), such that the following hold: Assuming a presentation of first-order logic without function symbols, the operator
models existential quantification over variable
κ λ
models the equality of variables
Hence, reformulated using standard logical notations, the axioms read as A cylindric set algebra of dimension
κ λ
κ , λ < α
κ λ
∣ x ( κ ) = x ( λ ) }
[1] It necessarily validates the axioms C1–C7 of a cylindric algebra, with
, set complement for complement, empty set as 0,
The set X is called the base.
A representation of a cylindric algebra is an isomorphism from that algebra to a cylindric set algebra.
Not every cylindric algebra has a representation as a cylindric set algebra.
[2][example needed] It is easier to connect the semantics of first-order predicate logic with cylindric set algebra.
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.
κ , λ
, the diagonals can be dropped out, and the following theorem of cylindric algebra (Pinter 1973): turns into the axiom of monadic Boolean algebra.
The axiom (C4) drops out (becomes a tautology).
Thus monadic Boolean algebra can be seen as a restriction of cylindric algebra to the one variable case.