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2-element Boolean algebra
Name: $\mathbb B_2=\langle\{0,1\},\vee,0,\wedge,1,'\rangle$
Elements: 0,1
Constant operations
0=0
1=1
Unary operations
| $x$ | 0 | 1 |
|---|---|---|
| $x'$ | 1 | 0 |
Binary operations
Join = or =
| $\vee$ | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 1 |
Meet = and = multiplication =
| $\wedge$ | 0 | 1 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 1 |
Derived operations
Symmetric difference: $x\oplus y=(x\vee y)\wedge(x\wedge y)'$
| $\oplus$ | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 0 |
Implication: $x\to y=x'\vee y$
| $\to$ | 0 | 1 |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 0 | 1 |
Bi-implication: $x\leftrightarrow y=(x\to y)\wedge(y\to x)$
| $\leftrightarrow$ | 0 | 1 |
|---|---|---|
| 0 | 1 | 0 |
| 1 | 0 | 1 |
Properties
| Simple | Yes |
|---|---|
| Subdirectly irreducible | Yes |
Bsaic results
This algebra generates the variety of all Boolean algebras.
Every Boolean algebra is a subdirect product of $\mathbb B_2$.
Maximal subalgebras
none
Minimal superalgebras
$\mathbb B_4$
Maximal homomorphic images
$\mathbb B_1$
Minimal homomorphic preimages
$\mathbb B_4$
Maximal subvarieties
Minimal supervarieties
???
Trace: » 2-element_boolean_algebra
