Cylindric algebra
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.
Contents
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:
(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
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.
See also
- 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
References
- 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.
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Further reading
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