2-element_boolean_algebra - MathStructures

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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|
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===

[[One-element algebras]]

===Minimal supervarieties===

???

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