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Concept
Algèbre cylindrique
Résumé
In mathematics, 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. 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 (where is any ordinal number) is an algebraic structure such that is a Boolean algebra, a unary operator on for every (called a cylindrification), and a distinguished element of for every and (called a diagonal), 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 . Hence, 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
A cylindric set algebra of dimension is an algebraic structure such that is a field of sets, is given by , and is given by . It necessarily validates the axioms C1–C7 of a cylindric algebra, with instead of , instead of , set complement for complement, empty set as 0, as the unit, and instead of . 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. It is easier to connect the semantics of first-order predicate logic with cylindric set algebra. (For more details, see .)
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.
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