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. 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
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:
(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
(C6) If is a variable different from both and , then
(C7) If and are different variables, then
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.
- Abstract algebraic logic
- Lambda calculus and Combinatory logic, other approaches to modelling quantification and eliminating variables
- Hyperdoctrines are a categorical formulation of cylindric algebras
- Relation algebras (RA)
- Polyadic algebra
- 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.
- Carlos Caleiro, Ricardo Gonçalves (2006). "On the algebraization of many-sorted logics". In J. Fiadeiro and P.-Y. Schobbens. Proc. 18th int. conf. on Recent trends in algebraic development techniques (WADT) (PDF). LNCS 4409. Springer. pp. 21–36. ISBN 978-3-540-71997-7.