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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

Complement = negation = $1-x$ =

$x$ | 0 | 1 |
---|---|---|

$x'$ | 1 | 0 |

#### 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 | 0 | 1 |

Nor: $x\downarrow y=(x\vee y)'$

$\downarrow$ | 0 | 1 |
---|---|---|

0 | 1 | 1 |

1 | 0 | 1 |

#### 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

$\mathbb B_2^2$

#### Maximal homomorphic images

$\mathbb B_1$

#### Minimal homomorphic preimages

#### Maximal subvarieties

#### Minimal supervarieties

???

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