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2-element_boolean_algebra [2010/08/16 10:41]
jipsen
2-element_boolean_algebra [2010/08/16 11:03] (current)
jipsen
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===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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===Derived operations=== ===Derived operations===
-Symmetric difference: $x\oplus y=(x\vee y)\wedge-(x\wedge y)$ +Symmetric difference: $x\oplus y=(x\vee y)\wedge(x\wedge y)'$ = $(x\wedge y')\vee(y\wedge x')$ 
-^$\vee$^0^1|+^$\oplus$^0^1|
^0|0|1| ^0|0|1|
^1|1|0| ^1|1|0|
-Implication: $x\to y=-x\vee y$ +Implication: $x\to y=x'\vee y$ 
-^$\vee$^0^1|+^$\to$^0^1|
^0|1|1| ^0|1|1|
^1|0|1| ^1|0|1|
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^0|1|0| ^0|1|0|
^1|0|1| ^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===
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^Subdirectly irreducible  |Yes  | ^Subdirectly irreducible  |Yes  |
-===Bsaic results===+===Basic results===
This algebra generates the variety of all [[Boolean algebras]]. This algebra generates the variety of all [[Boolean algebras]].
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===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===