Cylindric algebras

Abbreviation: CA$_\alpha$


A cylindric algebra of dimension $\alpha$ is a Boolean algebra with operators $\mathbf{A}=\langle A, \vee, 0, \wedge, 1, -, c_i, d_{ij}: i,j<\alpha\rangle$ such that for all $i,j<\alpha$

the $c_i$ are increasing: $x\le c_i x$

the $c_i$ semi-distribute over $\wedge$: $c_i(x\wedge c_i y) = c_i x\wedge c_i y$

the $c_i$ commute: $c_ic_j x=c_jc_i x$

the diagonals $d_{ii}$ equal the top element: $d_{ii}=1$

$d_{ij}=c_k(d_{ik}\wedge d_{kj})$ for $k\ne i,j$

$c_i(d_{ij}\wedge x)\wedge c_i(d_{ij}\wedge -x)=0$ for $i\ne j$

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.


Let $\mathbf{A}$ and $\mathbf{B}$ be … . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x ... y)=h(x) ... h(y)$


An is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$


Example 1:

Basic results


Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype variety
Equational theory undecidable for $\alpha\ge 3$, decidable otherwise
Quasiequational theory
First-order theory
Locally finite no
Residual size unbounded
Congruence distributive yes
Congruence modular yes
Congruence $n$-permutable yes, $n=2$
Congruence regular yes
Congruence uniform yes
Congruence extension property yes
Definable principal congruences
Equationally def. pr. cong.
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$


[[Representable cylindric algebras]] subvariety


[[Diagonal free cylindric algebras]] subreduct
[[Two-dimensional cylindric algebras]] subreduct