# Differences

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boolean_algebras [2010/07/29 18:30]
127.0.0.1 external edit
boolean_algebras [2010/09/04 17:13] (current)
jipsen
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=====Boolean algebras===== =====Boolean algebras=====
-Abbreviation: **BA**+Abbreviation: **BA** nbsp nbsp nbsp nbsp nbsp Search: [[http://www.google.com/search?q=boolean+algebras|Boolean algebras]]
====Definition==== ====Definition====
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==Morphisms== ==Morphisms==
-Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:Aarrow B$ that is a homomorphism: +Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(-x)=-h(x)$ $h(x\vee y)=h(x)\vee h(y)$, $h(-x)=-h(x)$
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====Definition==== ====Definition====
A \emph{Boolean algebra} is a [[Heyting algebra]] $\mathbf{A}=\langle A \emph{Boolean algebra} is a [[Heyting algebra]]$\mathbf{A}=\langle
-A,\vee ,0,\wedge ,1,arrow \rangle $such that+A,\vee ,0,\wedge ,1,\to\rangle$ such that
-$xarrow 0$ is an involution:  $( xarrow 0) arrow 0=x$+$\to 0$ is an involution:  $(x\to 0)\to 0=x$
====Examples==== ====Examples====
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0 & \text{otherwise}\end{array}. $0 & \text{otherwise}\end{array}.$
-\hyperbaseurl{http://math.chapman.edu/structures/files/}+
====Subclasses==== ====Subclasses====
[[One-element algebras]] [[One-element algebras]]
[[Complete Boolean algebras]] [[Complete Boolean algebras]]
+
====Superclasses==== ====Superclasses====
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====References==== ====References====
-[(Ln19xx>
-)]