Mathematical logic and foundations Definition

Cylindric algebra

Builds on Boolean algebra Ordinal number

Definition

Let $\alpha$ be an ordinal. A cylindric algebra of dimension $\alpha$ is a structure $(A, +, \cdot, -, 0, 1, c_\kappa, d_{\kappa\lambda})_{\kappa, \lambda < \alpha}$ with the following properties. The reduct $(A, +, \cdot, -, 0, 1)$ is a Boolean algebra, each $c_\kappa$ is a unary operation on $A$ , and each $d_{\kappa\lambda}$ is an element of $A$ . For all $x, y \in A$ and $\kappa, \lambda, \mu < \alpha$ :

$c_\kappa 0 = 0$ .
$x \le c_\kappa x$ .
$c_\kappa (x \cdot c_\kappa y) = c_\kappa x \cdot c_\kappa y$ .
$c_\kappa c_\lambda x = c_\lambda c_\kappa x$ .
$d_{\kappa\kappa} = 1$ .
If $\kappa \ne \lambda$ and $\kappa \ne \mu$ , then $d_{\lambda\mu} = c_\kappa (d_{\lambda\kappa} \cdot d_{\kappa\mu})$ .
If $\kappa \ne \lambda$ , then $c_\kappa (d_{\kappa\lambda} \cdot x) \cdot c_\kappa (d_{\kappa\lambda} \cdot -x) = 0$ .

Here $\le$ is the order of the Boolean algebra.

Remarks

Example: let $U$ be a nonempty set and let $A$ be the set of all subsets of $U^\alpha$ , with the set operations. Put $c_\kappa X = \set{y \in U^\alpha : y \text{ agrees with some } x \in X \text{ outside the coordinate } \kappa}$ and $d_{\kappa\lambda} = \set{x \in U^\alpha : x_\kappa = x_\lambda}$ . For clause 7, a point $y$ in both sets would agree outside $\kappa$ with some $x \in X$ and some $x' \notin X$ , both with $\kappa$ -th coordinate equal to $y_\lambda$ ; then $x = x'$ , a contradiction. The other clauses are checked in the same way, and this is a cylindric set algebra.

Non-example: on a Boolean algebra with $0 \ne 1$ , taking $c_\kappa x = 0$ for all $x$ fails clause 2 at $x = 1$ .

A cylindric algebra is called representable if it is isomorphic to a subalgebra of a product of cylindric set algebras. For dimension at least $3$ , not every cylindric algebra is representable.

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Boolean algebra concept
Ordinal number concept
Cylindric algebra definition

Tag	`RKCSWWVYTC`
URI	`m2://RKCSWWVYTC`
Kind	definition
Writing	W0 stub
MSC	03G15
Defines	Cylindric algebra
Uses	Boolean algebra, Ordinal number
Changed	2 November 2026

Cylindric algebra

Definition

Remarks

Reading path

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