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2-element Boolean algebra

Name: $\mathbb B_2=\langle\{0,1\},\vee,\wedge,',0,1\rangle$

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: $x\oplus y=(x\vee y)\wedge-(x\wedge y)$

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

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

Properties
Simple Yes
Subdirectly irreducible Yes
Notes

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

???