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2-element_boolean_algebra [2010/08/13 20:11]
jipsen created
2-element_boolean_algebra [2010/08/16 11:03] (current)
jipsen
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=====2-element Boolean algebra===== =====2-element Boolean algebra=====
-Name: $\mathbb B_2$+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
 +
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 = truncated addition = $\min\{x+y,1\}$ =
^$\vee$^0^1| ^$\vee$^0^1|
^0|0|1| ^0|0|1|
^1|1|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===
 +
 +???