Table of Contents
Boolean algebras
Abbreviation: BA nbsp nbsp nbsp nbsp nbsp Search: Boolean algebras Boolean rings
Definition
A \emph{Boolean algebra} is a structure $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,-\rangle $ of type $\langle 2,0,2,0,1\rangle $ such that
$0,1$ are identities for $\vee,\wedge$: $x\vee 0=x$, $x\wedge 1=x$
$-$ gives a complement: $x\wedge -x=0$, $x\vee -x=1$
$\vee,\wedge$ are associative: $x\vee (y\vee z)=(x\vee y)\vee z$, $x\wedge (y\wedge z)=(x\wedge y)\wedge z$
$\vee,\wedge$ are commutative: $x\vee y=y\vee x$, $x\wedge y=y\wedge x$
$\vee,\wedge$ are mutually distributive: $x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z)$, $x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z)$
Definition
A \emph{Boolean algebra} is a structure $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,-\rangle $ of type $\langle 2,0,2,0,1\rangle $ such that
$\langle A,\vee ,0,\wedge ,1\rangle $ is a bounded distributive lattice
$-$ gives a complement: $x\wedge -x=0$, $x\vee -x=1$
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(-x)=-h(x)$
It follows that $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(1)=1$.
Definition
A \emph{Boolean ring} is a structure $\mathbf{A}=\langle A,+ ,0,\cdot ,1\rangle $ of type $\langle 2,0,2,0\rangle $ such that
$\langle A,+ ,0,\cdot ,1\rangle $ is a commutative ring with unit
$\cdot$ is idempotent: $x\cdot x=x$
Remark: The term-equivalence with Boolean algebras is given by $x\wedge y=x\cdot y$, $-x=x+1$, $x\vee y=-(-x\wedge -y)$ and $x+y=(x\vee y)\wedge -(x\wedge y)$.
Definition
A \emph{Boolean algebra} is a Heyting algebra $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,\to\rangle $ such that
$\to 0$ is an involution: $(x\to 0)\to 0=x$
Examples
Example 1: $\langle \mathcal P(S), \cup ,\emptyset, \cap, S, -\rangle$, the collection of subsets of a sets $S$, with union, intersection, and setcomplementation.
Basic results
Properties
| Classtype | variety |
|---|---|
| Equational theory | decidable in NPTIME |
| Quasiequational theory | decidable |
| First-order theory | decidable |
| 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 | yes |
| Equationally def. pr. cong. | yes |
| Amalgamation property | yes |
| Strong amalgamation property | yes |
| Epimorphisms are surjective | yes |
| Locally finite | yes |
| Residual size | 2 |
Finite members
Number of algebras $=\{
\begin{array}{cc}
1 & \text{if size}=2^{n}
0 & \text{otherwise}\end{array}. $
