Differences

This shows you the differences between two versions of the page.

2-element_boolean_algebra [2010/08/16 10:45]
jipsen
2-element_boolean_algebra [2010/08/16 11:03] (current)
jipsen
Line 15: Line 15:
^$x$ |0|1| ^$x$ |0|1|
^$x'$|1|0| ^$x'$|1|0|
 +Alternative notation: $-x=\overline x=x^-=\neg x$
===Binary operations=== ===Binary operations===
Line 29: Line 30:
===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| ^$\oplus$^0^1|
^0|0|1| ^0|0|1|
Line 43: Line 44:
^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===
Line 48: Line 59:
^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]].
Line 59: Line 70:
===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===