Table of Contents
Boolean algebras with operators
Abbreviation: BAO
Definition
A \emph{Boolean algebra with operators} is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,f_i\ (i\in I)\rangle$ such that
$\langle A,\vee,0,\wedge,1,\neg\rangle$ is a Boolean algebra
$f_i$ is \emph{join-preserving} in each argument: $f_i(\ldots,x\vee y,\ldots)=f_i(\ldots,x,\ldots)\vee f_i(\ldots,y,\ldots)$
$f_i$ is \emph{normal} in each argument: $f_i(\ldots,0,\ldots)=0$
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean algebras with operators of the same signature. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves all the operators:
$h(f_i(x_0,\ldots,x_{n-1}))=f_i(h(x_0),\ldots,h(x_{n-1}))$
Examples
Example 1:
Basic results
Properties
| Classtype | variety |
|---|---|
| Equational theory | decidable |
| Quasiequational theory | |
| First-order theory | undecidable |
| 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 | no |
| Equationally def. pr. cong. | no |
| Amalgamation property | yes |
| Strong amalgamation property | yes |
| Epimorphisms are surjective | yes |
