# Differences

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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===
+
+???