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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 = $1-x$ =

$x$ 01
$x'$10

Alternative notation: $-x=\overline x=x^-=\neg x$

Binary operations

Join = or = truncated addition = $\min\{x+y,1\}$ =

$\vee$01
001
111

Meet = and = multiplication =

$\wedge$01
000
101

Derived operations

Symmetric difference: $x\oplus y=(x\vee y)\wedge(x\wedge y)'$ = $(x\wedge y')\vee(y\wedge x')$

$\oplus$01
001
110

Implication: $x\to y=x'\vee y$

$\to$01
011
101

Bi-implication: $x\leftrightarrow y=(x\to y)\wedge(y\to x)$

$\leftrightarrow$01
010
101

Nand: $x|y=(x\wedge y)'$

$|$01
011
101

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

$\downarrow$01
011
101

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

Minimal homomorphic preimages

4-element Boolean algebra $\mathbb B_2^2$

Maximal subvarieties

Minimal supervarieties

???