Cylindric algebra

In mathematics, the notion of cylindric algebra, developed 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.

The cylindric algebra should not be confused with the measure theoretic concept cylindrical algebra that arises in the study of cylinder set measures and the cylindrical σ-algebra.

A cylindric algebra of dimension ${\displaystyle \alpha }$ (where ${\displaystyle \alpha }$ is any ordinal number) is an algebraic structure ${\displaystyle (A,+,\cdot ,-,0,1,c_{\kappa },d_{\kappa \lambda })_{\kappa ,\lambda <\alpha }}$ such that ${\displaystyle (A,+,\cdot ,-,0,1)}$ is a Boolean algebra, ${\displaystyle c_{\kappa }}$ a unary operator on ${\displaystyle A}$ for every ${\displaystyle \kappa }$ (called a cylindrification), and ${\displaystyle d_{\kappa \lambda }}$ a distinguished element of ${\displaystyle A}$ for every ${\displaystyle \kappa }$ and ${\displaystyle \lambda }$ (called a diagonal), such that the following hold:

(C1) ${\displaystyle c_{\kappa }0=0}$

(C2) ${\displaystyle x\leq c_{\kappa }x}$

(C3) ${\displaystyle c_{\kappa }(x\cdot c_{\kappa }y)=c_{\kappa }x\cdot c_{\kappa }y}$

(C4) ${\displaystyle c_{\kappa }c_{\lambda }x=c_{\lambda }c_{\kappa }x}$

(C5) ${\displaystyle d_{\kappa \kappa }=1}$

(C6) If ${\displaystyle \kappa \notin \{\lambda ,\mu \}}$, then ${\displaystyle d_{\lambda \mu }=c_{\kappa }(d_{\lambda \kappa }\cdot d_{\kappa \mu })}$

(C7) If ${\displaystyle \kappa \neq \lambda }$, then ${\displaystyle c_{\kappa }(d_{\kappa \lambda }\cdot x)\cdot c_{\kappa }(d_{\kappa \lambda }\cdot -x)=0}$

Assuming a presentation of first-order logic without function symbols, the operator ${\displaystyle c_{\kappa }x}$ models existential quantification over variable ${\displaystyle \kappa }$ in formula ${\displaystyle x}$ while the operator ${\displaystyle d_{\kappa \lambda }}$ models the equality of variables ${\displaystyle \kappa }$ and ${\displaystyle \lambda }$. Hence, reformulated using standard logical notations, the axioms read as

(C1) ${\displaystyle \exists \kappa .{\mathit {false}}\iff {\mathit {false}}}$

(C2) ${\displaystyle x\implies \exists \kappa .x}$

(C3) ${\displaystyle \exists \kappa .(x\wedge \exists \kappa .y)\iff (\exists \kappa .x)\wedge (\exists \kappa .y)}$

(C4) ${\displaystyle \exists \kappa \exists \lambda .x\iff \exists \lambda \exists \kappa .x}$

(C5) ${\displaystyle \kappa =\kappa \iff {\mathit {true}}}$

(C6) If ${\displaystyle \kappa }$ is a variable different from both ${\displaystyle \lambda }$ and ${\displaystyle \mu }$, then ${\displaystyle \lambda =\mu \iff \exists \kappa .(\lambda =\kappa \wedge \kappa =\mu )}$

(C7) If ${\displaystyle \kappa }$ and ${\displaystyle \lambda }$ are different variables, then ${\displaystyle \exists \kappa .(\kappa =\lambda \wedge x)\wedge \exists \kappa .(\kappa =\lambda \wedge \neg x)\iff {\mathit {false}}}$

A cylindric set algebra of dimension ${\displaystyle \alpha }$ is an algebraic structure ${\displaystyle (A,\cup ,\cap ,-,\emptyset ,X^{\alpha },c_{\kappa },d_{\kappa \lambda })_{\kappa ,\lambda <\alpha }}$ such that ${\displaystyle \langle X^{\alpha },A\rangle }$ is a field of sets, ${\displaystyle c_{\kappa }S}$ is given by ${\displaystyle \{y\in X^{\alpha }\mid \exists x\in S\ \forall \beta \neq \kappa \ y(\beta )=x(\beta )\}}$, and ${\displaystyle d_{\kappa \lambda }}$ is given by ${\displaystyle \{x\in X^{\alpha }\mid x(\kappa )=x(\lambda )\}}$.^[1] It necessarily validates the axioms C1–C7 of a cylindric algebra, with ${\displaystyle \cup }$ instead of ${\displaystyle +}$, ${\displaystyle \cap }$ instead of ${\displaystyle \cdot }$, set complement for complement, empty set as 0, ${\displaystyle X^{\alpha }}$ as the unit, and ${\displaystyle \subseteq }$ instead of ${\displaystyle \leq }$. 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.^{[2][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.

When ${\displaystyle \alpha =1}$ and ${\displaystyle \kappa ,\lambda }$ are restricted to being only 0, then ${\displaystyle c_{\kappa }}$ becomes ${\displaystyle \exists }$, the diagonals can be dropped out, and the following theorem of cylindric algebra (Pinter 1973):

${\displaystyle c_{\kappa }(x+y)=c_{\kappa }x+c_{\kappa }y}$

turns into the axiom

${\displaystyle \exists (x+y)=\exists x+\exists y}$

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

• 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