5.2 Cylindric algebras: definition and basic properties
2 min read•july 24, 2024
Cylindric algebras expand on Boolean algebras, adding cylindrification operators and diagonal elements. These additions allow for more complex logical structures, mirroring existential quantification and equality in first-order logic.
Key properties of cylindric algebras include expansion, distribution, and idempotence of cylindrification operators. These properties, along with the interaction between cylindrification and Boolean operations, form the foundation for working with these advanced algebraic structures.
Cylindric Algebras: Foundations and Properties
Definition of cylindric algebras
Top images from around the web for Definition of cylindric algebras
Boolean algebra (structure) - Wikipedia View original
Is this image relevant?
1 of 2
Cylindric algebras extend Boolean algebras with additional operations and elements
Basic components combine structure with cylindrification operators ci for each dimension i and diagonal elements dij for dimension pairs i and j
Cylindrification operators ci generalize existential quantification by projecting elements onto higher dimensions
Diagonal elements dij represent equality between variables in different dimensions (x = y)
Inherited Boolean operations include join (∨), meet (∧), complement (¬)
Specific axioms govern cylindric algebras such as commutativity of cylindrification (cicjx=cjcix) and cylindrification of diagonal elements (cidij=1)
Properties of cylindric algebras
Cylindrification axioms describe fundamental behavior of ci operators:
Expansion: x≤cix
Distribution: ci(x∧ciy)=cix∧ciy
Idempotence: cicix=cix
Proof techniques utilize Boolean algebra laws, -specific axioms, and induction on term structure
Key properties include monotonicity of cylindrification, interaction with Boolean operations, and relationship between cylindrification and diagonal elements
Monotonicity states if x≤y, then cix≤ciy
Interaction with Boolean operations shows ci(x∨y)=cix∨ciy
Relationship between cylindrification and diagonal elements demonstrates ci(dij∧x)=x if i=j
Cylindric vs Boolean algebras
Cylindric algebras extend Boolean algebras while retaining all Boolean operations and laws
Dimensions in cylindric algebras correspond to number of free variables (Boolean algebras are 0-dimensional)
Representation theory extends Stone's theorem for Boolean algebras to cylindric algebras
Algebraization of logic shows Boolean algebras algebraize propositional logic while cylindric algebras algebraize first-order logic with equality
Construction of cylindric algebras
Finite cylindric algebras based on power sets of finite sets with dimension determined by coordinate number in base set
Set-theoretic cylindric algebras use power set of ωX (functions from ω to X) as universe
Cylindrification in set-theoretic algebras defined as ci(A)={s∈ωX:∃t∈A,t(j)=s(j)forj=i}
Diagonal elements in set-theoretic algebras defined as dij={s∈ωX:s(i)=s(j)}
Cylindric algebras from relational structures use set of all formulas in given language
Dimension determined by counting distinct cylindrification operators or analyzing diagonal element structure
Concrete examples include 2-dimensional cylindric algebra of binary relations and 3-dimensional cylindric algebra of ternary relations