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A boolean algebra is a LatticE which satisfies the following properties:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c)                                     (distributive law)
    For all a, there exists an element ~a such that av~a=1 and a^~a=0      (existence of complements)

Complements are guaranteed to be unique within bounded distributive lattices. Note the definition of Boolean algebrae is very similar to that of RinGs, except elements have complements instead of inverses. Moreover the distributive law can be shown to hold both ways, i.e. (a^b)vc=(avc)^(bvc).

The PowerSet? of any given set S forms a boolean algebra under the partial ordering "is a SubSet of", where 0={} and 1=S. Any subalgebra of this is called an algebra of sets, and in particular, an algebra of sets closed under arbitrary unions is called a TopOlogy.

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Last edited January 29, 2001 9:01 am by JoshuaGrosse (diff)