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
Complement = negation = $1-x$ =
| $x$ | 0 | 1 |
|---|---|---|
| $x'$ | 1 | 0 |
Alternative notation: $-x=\overline x=x^-=\neg x$
Binary operations
Join = or = truncated addition = $\min\{x+y,1\}$ =
| $\vee$ | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 1 |
Meet = and = multiplication =
| $\wedge$ | 0 | 1 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 1 |
Derived operations
Symmetric difference: $x\oplus y=(x\vee y)\wedge(x\wedge y)'$ = $(x\wedge y')\vee(y\wedge x')$
| $\oplus$ | 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 |
Nand: $x|y=(x\wedge y)'$
| $|$ | 0 | 1 |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1 | 0 |
Nor: $x\downarrow y=(x\vee y)'$
| $\downarrow$ | 0 | 1 |
|---|---|---|
| 0 | 1 | 0 |
| 1 | 0 | 0 |
Properties
| Simple | Yes |
|---|---|
| Subdirectly irreducible | Yes |
Basic 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
4-element Boolean algebra $\mathbb B_2^2$
Maximal homomorphic images
1-element Boolean algebra $\mathbb B_1$
Minimal homomorphic preimages
4-element Boolean algebra $\mathbb B_2^2$
Maximal subvarieties
Minimal supervarieties
???
