Cylindric algebra - AbsoluteAstronomy.com

The notion of cylindric algebra, invented by Alfred Tarski

Alfred Tarski

Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...

, arises naturally in the algebraization

Algebraic logic

In mathematical logic, algebraic logic is the study of logic presented in an algebraic style.What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics and connected problems...

of first-order logic

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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

Quantification

Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method.In logic,...

and equality. They differ from polyadic algebra

Polyadic algebra

Polyadic algebras are algebraic structures introduced by Paul Halmos. They are related to first-order logic in a way analogous to the relationship between Boolean algebras and propositional logic .There are other ways to relate first-order logic to algebra, including Tarski's cylindric algebras...

s in that the latter do not model equality.

Definition of a cylindric algebra

A cylindric algebra of dimension

, where

is any ordinal

Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...

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

Existential quantification

In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. It is denoted by the logical operator symbol ∃ , which is called the existential quantifier...

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

, then

(C7) If

, then

Generalizations

Recently, cylindric algebras have been generalized to the many-sorted

Many-sorted logic

Many-sorted logic can reflect formally our intention, not to handle the universe as a homogeneous collection of objects, but to partition it in a way that is similar to types in typeful programming...

case, which allows for a better modeling of the duality between first-order formulas and terms.

External links

Jipsen's algebra page.

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Definition of a cylindric algebra

Generalizations

See also

External links