## Cylindric algebras

Abbreviation: **CA$_\alpha$**

### Definition

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 $i\ne j$

Remark: This is a template. Click on the 'Edit text of this page' link at the bottom to add some information to this page.

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.

##### Morphisms

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: $h(x ... y)=h(x) ... h(y)$

### Definition

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

$...$ is …: $axiom$

$...$ is …: $axiom$

### Examples

Example 1:

### Basic results

### Properties

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

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

### Subclasses

`[[Representable cylindric algebras]] subvariety`

### Superclasses

[[Diagonal free cylindric algebras]] subreduct

[[Two-dimensional cylindric algebras]] subreduct

### References

Trace: » cylindric_algebras