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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

$x$ 01
$x'$10

Binary operations

Join = or =

$\vee$01
001
111

Meet = and = multiplication =

$\wedge$01
000
101

Derived operations

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

$\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

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

???