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2-element_boolean_algebra [2010/08/16 10:52]
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2-element_boolean_algebra [2010/08/16 11:03] (current)
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^$x$ |0|1| ^$x$ |0|1|
^$x'$|1|0| ^$x'$|1|0|
 +Alternative notation: $-x=\overline x=x^-=\neg x$
===Binary operations=== ===Binary operations===
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^$|$^0^1| ^$|$^0^1|
^0|1|1| ^0|1|1|
-^1|0|1|+^1|1|0|
Nor: $x\downarrow y=(x\vee y)'$ Nor: $x\downarrow y=(x\vee y)'$
^$\downarrow$^0^1| ^$\downarrow$^0^1|
-^0|1|1+^0|1|0
-^1|0|1|+^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===