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.
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 (called a cylindrification), and a distinguished element of for every and (called a diagonal), 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 . Hence, 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
Cylindric set algebras
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.[example needed] It is easier to connect the semantics of first-order predicate logic with cylindric set algebra. (For more details, see § Further reading.)
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.
Relation to monadic Boolean algebra
When and are restricted to being only 0, then becomes , 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.
- 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
- Cylindrical algebraic decomposition
- Hirsch and Hodkinson p167, Definition 5.16
- Hirsch and Hodkinson p168
- Charles Pinter (1973). "A Simple Algebra of First Order Logic". Notre Dame Journal of Formal Logic. XIV: 361–366.
- Leon Henkin, J. Donald Monk, and Alfred Tarski (1971) Cylindric Algebras, Part I. North-Holland. ISBN 978-0-7204-2043-2.
- Leon Henkin, J. Donald Monk, and Alfred Tarski (1985) Cylindric Algebras, Part II. North-Holland.
- Robin Hirsch and Ian Hodkinson (2002) Relation algebras by games Studies in logic and the foundations of mathematics, North-Holland
- Carlos Caleiro, Ricardo Gonçalves (2006). "On the algebraization of many-sorted logics" (PDF). In J. Fiadeiro and P.-Y. Schobbens (ed.). Proc. 18th int. conf. on Recent trends in algebraic development techniques (WADT). LNCS. Vol. 4409. Springer. pp. 21–36. ISBN 978-3-540-71997-7.
- 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.
- example of cylindrical algebra by CWoo on planetmath.org