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2-element_boolean_algebra [2010/08/16 10:34]
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2-element_boolean_algebra [2010/08/16 11:03] (current)
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=====2-element Boolean algebra===== =====2-element Boolean algebra=====
-Name: $\mathbb B_2=\langle\{0,1\},\vee,\wedge,',0,1\rangle$+Name: $\mathbb B_2=\langle\{0,1\},\vee,0,\wedge,1,'\rangle$
Elements: 0,1 Elements: 0,1
-==Constant operations==+===Constant operations===
0=0 0=0
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1=1 1=1
-==Unary operations==+===Unary operations=== 
 +Complement = negation = $1-x$ =
^$x$ |0|1| ^$x$ |0|1|
^$x'$|1|0| ^$x'$|1|0|
 +Alternative notation: $-x=\overline x=x^-=\neg x$
-==Binary operations== +===Binary operations=== 
-Join = or =+Join = or = truncated addition = $\min\{x+y,1\}$ =
^$\vee$^0^1| ^$\vee$^0^1|
^0|0|1| ^0|0|1|
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^1|0|1| ^1|0|1|
-==Derived operations==+===Derived operations===
-Symmetric difference: $x\oplus y=(x\vee y)\wedge-(x\wedge y)$+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: $x\to y=-x\vee y$+Implication: $x\to y=x'\vee y$ 
 +^$\to$^0^1| 
 +^0|1|1| 
 +^1|0|1|
Bi-implication: $x\leftrightarrow y=(x\to y)\wedge(y\to x)$ 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==+===Properties===
^Simple  |Yes  | ^Simple  |Yes  |
^Subdirectly irreducible  |Yes  | ^Subdirectly irreducible  |Yes  |
-==Notes==+===Basic results===
This algebra generates the variety of all [[Boolean algebras]]. This algebra generates the variety of all [[Boolean algebras]].
Every Boolean algebra is a subdirect product of $\mathbb B_2$. Every Boolean algebra is a subdirect product of $\mathbb B_2$.
-==Maximal subalgebras==+===Maximal subalgebras===
none none
-==Minimal superalgebras==+===Minimal superalgebras===
-$\mathbb B_4$+[[4-element Boolean algebra]] $\mathbb B_2^2$
-==Maximal homomorphic images==+===Maximal homomorphic images===
-$\mathbb B_1$+[[1-element Boolean algebra]] $\mathbb B_1$
-==Minimal homomorphic preimages==+===Minimal homomorphic preimages===
-$\mathbb B_4$+[[4-element Boolean algebra]] $\mathbb B_2^2$
-==Maximal subvarieties==+===Maximal subvarieties===
[[One-element algebras]] [[One-element algebras]]
-==Minimal supervarieties==+===Minimal supervarieties===
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